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THE MEASIREMEXT 



OF 



GE'^V^ERAL EXCHAXGE-YALUE 



BY 

CORREA MOYLAN WALSH 



THE MACMILLAX COMPANY 

LONDON: MACMILLAN & CO., Ltd. 
1901 



THE LIBRARY OF 

CONGRESS, 
Two Copies Received 

MAY. 7 1901 

Copyright entry 

CLASS Cl^XXa N«. 

COPY 3. 



Copyright, 1901, 
By the MACMILLAN COMPANY. 



PRESS OF 

THE NEW ERA PRINTING COMPANY, 

LANCASTER. PA, 



CONTENTS. 



CHAPTER I. 

THE NATURE OF EXCHANGE-VALUE. 

Section I. Four kinds of economic value.- 'i 1. The four kinds explained. 
^ 2. Their common characteristics and differentise. | 3. Confusion of 
opinion on the subject. ....... pp. 1-6 

Section II. The relativity of exchange-value. ^ 1. Exchange-value, like 
gravity, a relative quality and power in tilings. Its causes, partly resid- 
ing in men, immaterial in mensuration. | 2. Denial of exchange-value as 
a quality in things incorrect. | 3. Particular exchange-value ; not the 
quantity of another thing, nor a mere relation. | 4. General exchange- 
value : two kinds. ........ pp. 7-13 

Section III. The quantitativeness of exchange-value. ? 1. Variability of 
exchange-value — local and temporal. | 2. Measurement of particular 
exchange -values. § 3. Equality between exchange-values. ? 4. Prac- 
tical difficulties in determining jiarticular exchange-values. § 5. Prac- 
tical difficulties in determining general exchange-values ; assistance 
rendered by money. § 6. Measurement of general exchange-value in- 
dispensable pp. 14-22 

Section IV. Mensuration not concerned with causes. § 1. Causes to be in- 
vestigated after the measurement of variations. ^ 2. Errors of economists 
on the subject. | 3. Mensuration of exchange-value not useless because 
different from mensuration of other kinds of value. . . pp. 22-25 

CHAPTER II. 

THE CORRELATION OF EXCHANGE-VALUES. 

Section I. The correlation of the exchange- values of two things. ^1. The 
variation of one thing in another involves an inverse variation of the 
other in it. ^ 2. The mathematical relations in such variations — recipro- 
cal ratios. I 3. The equality of such variations, although the propor- 
tions are different. ^ 4. Accompanying relations to other things. 

pp. 26-32 
V 



VI CONTENTS 

Section II. The correlation between the exchange-value of one thing and 
the exchange-value of all others. § 1. The inverse variations of partic- 
ular exchange-values involved ; and the measurement of the variation of 
the one thing in exchange-value in all the others. § 2. The variations 
of the others in general exchange-value are smaller than the variations of 
the one. § 3. The same principles stated in terms of money. 

pp. 32-36 

Section III. Impossibility of one thing alone varying in exchange-value. 

^ 1. The principle stated ; confined to exchange-value. § 2. Error from 
not recognizing it ; needlessness of reasoning from probability. 

pp. 36-39 

Section IV. Exchange-value in all other things and exchange-value in all 
things. § 1. The distinction explained. | 2. Their differences. ^ 2. 
Their identity in certain cases. ..... pp. 39-43 

Section V. Possibility of constancy of general exchange-value. ^ 1. Two 
conditions. | 2. Denial of it under variations of particular exchange- 
values ; consequences. ^ 3. Refutation ; and demonstration of the pos- 
sibility — by compensation in opposite variations. | 4. Relations between 
things constant and things that vary. § 5. Collectively all the other 
things may be constant also. § 6. Summary. . . . pp. 44-53 



CHAPTEE III. 

ON THE MEASUEEMENT OF GENERAL EXCHANGE-VALUE. 

Section I. Comparison with other measurements. | 1. Seeming peculiar- 
ity of our subject because of the relativity of exchange-value. ^ 2. The 
difference really only of degree. ..... pp. 54-56 

Section II. The standard in simple mensuration. | 1. Need of a standard 
in every kind of mensuration. | 2. Two kinds of subjects of mensura- 
tion. I 3. In subjects of simple mensuration the standard is the relation 
between a whole and its parts, g 4. The fixity neither in the whole nor 
in the parts singly. ^ 5. We want our measures to be fixed parts re- 
latively to the universe. § 6. We practically content ourselves with 
reference to a smaller whole. | 7. And there is need of compensatory 
variations even in the mensuration of length and weight. § 8. The 
mensuration of exchange-value not essentially different. § 9. The term 
" absolute " can apply only in a secondary sense. . . pp. 56-67 

Section III. The distinction of two kinds not peculiar to general exchange- 
value. § 1. The distinction likewise in the cases of length and motion. 
^ 2. Probability excluded also here pp. 67-70 

Section IV. The true peculiarity in the mensuration of general exchange- 
value. I 1. The economic worlds compared are not the same or exactly 
alike. ^ 2. This difficulty in a less degree also in other subjects. § 3. 
Two ways of overcoming it. | 4. The amount of precision needed. 

pp. 70-75 



CONTENTS 



CHAPTER IV. 

SELECTION AND ARRANGEMENT OF PARTICULAR EXCHANGE-VALUES. 

Section I. The Selection, 'i 1. Need of curtailment. § 2. Classes only to 
be counted, and of these the simplest. ? 3. Still further restriction for 
measurements in the past. § 4. Only wholesale prices to be used. 

pp. 76-80 

Section II. Weighting, 'i 1. Definition of weighting. | 2. Haphazard and 
even weighting. ^ 3. Need of care in weighting. | 4. Various opinions 
about weighting ; whether it should be according to importance. •? 5. 
On weighting according to physical weight or bulk. . pp. 80-87 

Section III. Weighting examined and explained. ? 1. The nature of weight- 
ing. ^ 2. Problem about the individuals in classes. ^ 3. Economic in- 
dividuals not equal weights or bulks. § 4. They are equivalents. ^ 5. 
The size of the class the variation of which in exchange-value is being 
measured, indifferent. . . ..... pp. 87-95 

Section IV. Some details in weighting, 'i 1. Thn quantities not at a mo- 
ment, but during a period. ^ 2. The quantity produced or the quantity 
consumed, according to which is the larger. § 3. Materials in different 
stages of production. ^ 4. Weighting also in obtaining the price of each 
class during a given period. . . .... pp. 95-97 

Section V. The question of periods, 'i 1. Various opinions on the subject. 
§ 2. Three elemental cases. ^ 3. I. Classes with constant relative money- 
values at both periods, to be weighted according to these. § 4. II. Classes 
with constant relative mass-quantities at both periods. The weighting of 
neither period preferable. ? 5. Nor the weights that are common to both 
periods, being the smaller at either. § 6. A mean must be used, and this 
the geometric. ^ 7. An example. ^ 8. III. Classes with varying money- 
values and mass-qmintities. Various positions. § 9. On the appearance 
and disappearance of classes ; these not to be counted. ^ 10. But the 
individuals that appear or disappear in old classes are to be counted. 
Ol. Fui'ther reasons against taking only the lesser mass-quantities at 
either period. § 12. Nor is the arithmetic mean of the mass-quantities 
proper. § 13. But the geometric mean of the total money-values. ? 14. 
Less exactness required in measuring variations in the past. pp. 97-121 

Section VI. Exclusion of wages and earnings in general. § 1. The claim 
that the "price of labor" should be included. ^ 2. Labor, not an ex- 
changeable object, has not exchange-value. § 3. Labor the measure of 
cost-value, and in a way of esteem-value. ^ 4. Desirable also to measure 
the variations in the esteem-value of money. ^ 5. This by means of the 
average money-earnings of people. ^ 6. The measurement of the varia- 
tions in the exchange-value of money a help also in measuring variations 
in the esteem-value of all things in general. ? 7. Absurdity of mixing 
the two measurements, one of each of two kinds of value, of money. 
^ 8. It confuses also a standard for one stratum of society with a standard 



Vm CONTEXTS 

for all people. § 9. Difficulty about weighting wages and commodities 
relatively to each other. § 10. Neither the measurement of esteem-value 
. nor the measurement of exchange-value is more properly the measure- 
ment of "value." ^ 11. The measurements of the different kinds of 
value should be kept distinct. ..... pp. 121-135 



CHAPTER V. 

MATHEMATICAL FORMULATION OF EXCHANGE-VALUE RELATIONS. 

Section I. Formulation of exchange-value relations with even weighting. 

§ 1. On the need of measuring our measures. I 2. Preliminary nota- 
tion. § 3. Further notation, to express equations. ^ 4. Formulation of 
exchange-value in all other things, — three averages. § 5. Formulation 
of the variation of such general ex change- value, — still three averages. 
I 6. Another notation for the formulation of variations. ^ 7. Meaning 
of the formulae, and their differences. .... pp. 136-150 

Section II. Formulation of exchange-value relations with uneven weight- 
ing. ^ 1. The previous formulse modified. § 2. Formulae with double 
weighting pp. 150-156 

Section III. Warnings against possible errors. ^ 1. On the variation of 
averages and the average of variations. ^ 2. The two should agree. 
§^3, 4. Possible errors in arithmetic averaging — perverting even weight- 
ing ; perverting uneven weighting. § 5. Possible errors in harmonic 
averaging. ^ 6. Possible errors in geometric averaging. ^ 7. Another 
cause of such errors : false notation. § 8. On the notion of purchasing 
power pp. 157-173 

Section IV. Formulation of price relations with even weighting. § 1. 

Conversion of the preceding formulae into formulae for prices, and the 
relation between the averages. | 2. The simplest formulae for price 
variations. ......... pp. 173-178 

Section V. Formulation of price relations with uneven weighting. ^ 1. 
The last formula modified. I 2. Formulae with double weighting. 

pp. 178-184 

Section yi. Possible errors, and errors incurred, g 1. The criterion, that 
the variation of the averages should agree with the average of the varia- 
tions. § 2. Possible errors in harmonic averaging. ^^ 3, 4. Possible and 
actual errors in arithmetic averaging: — haphazard weighting in Dutot's 
and Carli' s methods ; in Scrope' s and Young' s methods, 'i 5. Error of 
Drobisch's method, and its true nature. I 6. Possible ei'rors in geometric 
averaging. § 7. An excellence in this averaging : Westergaard' s argu- 
ment for it ; remarks thereon. § 8. The proper procedure in a series of 
measurements pp. 184-208 

Section VII. Formulation of exchange-value in all things. ? 1. Formulae 

for averaging variations. ^ 2. Formulae for variations of averages. 

pp. 208-211 



CONTENTS IX 

CHAPTER VI. 

THE QUESTION OF THE MEANS AND AVERAGES. 

Section I. The problem and suggested solutions. ? 1. The problem in its 
simplest form. § 2. Some hasty solutions refuted. ^ 3. Difficulties in 
the subject. ? 4. The means and averages suggested as solutions. 

pp. 212-218 

Section II. History of 'the question. § 1. The unconscious adoption of the 
arithmetic average. ^ 2. The question raised by Jevons, who adopted 
the geometric average. § 3. General retention of the arithmetic aver- 
age. ^ 4. The liarmonic average of price variations, and others. § 5. 
Attempted rejection of all the averages, by employment of double weight- 
ing pp. 218-225 

CHAPTER VII. 

BRIEF COMPARISON OF THE MEANS. 

Section I. Another form of the same problem. | 1. Statement of the new 
form. ^ 2. Formulation of it. . . . . . pp. 226-228 

Section II. The comparison. ^ 1. Suggestion of the three answers. §2. 
A table of compensatory variations, showing their characteristics. § 3. ■ 
Excessiveness of the arithmetic and harmonic answei's ; moderation of 
the geometric pp. 228-232 

CHAPTER VIII. 

THE GENERAL ARGUMENT FOR THE GEOMETRIC MEAN. 

Section I. Equality in opposite variations ; representation of it by the 
means and averages. § 1. Three possible ways of conceiving of equality 
in opposite variations claiming to be compensatory. § 2. These ex- 
pounded. I 3. Equality of distance traversed. § 4. Formulation of 
these equalities ; and their conditions. | 5. These equalities exist in 
complex as well as in simple arithmetic and harmonic variations. § 6. 
But they do not exist in complex geometric variations. Diflferentiation 
of the geometric average from]]the geometric mean. § 7. Connection be- 
tween the means or averages and the periods of weighting. Equal op- 
posite variations, simply, are geometric. ? 8. In our subject the argu- 
ments for the geometric equality apply on both sides of the treatment of 
exchange-value pp. 233-245 

Section II. Natures of the subjects to which the means and averages are 
applicable. I 1. Diflerences in regard to the limits on the opposite sides. 
§ 2. Our subject of the kind for which the geometric mean is suitable. 

pp. 245-249 

Section III. The means and averages in connection with exchange-values 
and prices. ^1. Real inequality in the arithmetic and harmonic varia- 



X CONTENTS 

tions. § 2. Keal equality in the simple geometric variations. § 3. A 
statement by Jevons explained. § 4. The argument not applicable to 
the geometric average. ....... pp. 249-255 

CHAPTEE IX. 

REVIEW OF THE ARGUMENTS FOR THE HARMONIC AND ARITHMETIC 
AVERAGES OF PRICE VARIATIONS. 

Section I. Analysis of the Arguments. § 1. The argument for the harmonic 
average of price variations — from compensation by equal mass-quantities. 
§ 2. The argument for the arithmetic average of price variations — from 
compensation by equal sums of money. § 3. Comparison of these argu- 
ments. § 4. Faultiness of the argument for the harmonic average. 

pp. 256-262 

Section II. Mathematical relations ignored in these arguments. | 1. In- 
terconvertibility of the arguments ; their applicability also to the geo- 
metric mean. § 2. Apparently equal defectiveness of both the arguments, 
applied to any of the averages. | 3. Laspeyres's argument for the arith- 
metic average. | 4. Compensation by equal mass-quantities possible in 
all cases of opposite price variations. § 5. Compensation by equal sums 
of money also possible in all such cases. . . . pp. 263-274 

Section III. Correction of the defects in these arguments. | 1. Their gen- 
eral defect is neglect of weighting, f 2. Also, neglect concerning the 
mass-units ; effect of this upon the argument from compensation by equal 
mass-quantities. Formulation of this argument. § 3. Formulation of 
the argument from compensation by equal sums of money. § 4. The 
two arguments apply to different conditions, and so are not antagonistic. 

pp. 274-282 

CHAPTER X. 

THE METHOD FOR CONSTANT SUMS OF MONEY. 
Section I. Schematization of the argument from compensation by equal 
mass-quantities. § 1. The equality not in masses but in proportions, — 
three kinds. | 2. Schemata for mass-units equivalent at the first period. 
§ 3. Schemata for mass-units equivalent at the second period. § 4. 
Schemata for mass-units equivalent over both the periods, pp. 283-292 

Section II. What this argument really proves. §1. The appearance of cor- 
rectness in the argument as used by the harmonic averagist. | 2. The 
fallacy in the argument for the harmonic average of price variations. 
I 3. The fallacy in the argument applied to the arithmetic average of 
price variations. § 4. Correctness of the argument for the geometric 
mean of price variations. | 5. Eemarks on "purchasing power." 

pp. 293-305 

Section III. Formulation of the method. ^ 1 . The general principle, and 
the formulae in reduced mass-units. | 2. These formulje extended and 



CONTENTS XI 

interpreted, 'i 3. Formulation in ordinary mass-units, and reduction to 
a form of Scrope's method. § 4. The method satisfies all the Proposi- 
tions. ^ 5. Comparison of the formulif ; cases when the geometric aver- 
age may be right. ........ pp. 305-313 

Section IV. The deviation of the geometric average. | 1. Example with 
the more important class falling in price. ^ 2. Example with the more 
important class rising in price. ^ 3. Probable direction and extent of 
the error ; likelihood of neutralization in a long series. pp. 313-324 

Section V. Testing of the method in a series. § 1. Example with simple 
harmonic price variations. | 2. Example with simple arithmetic price 
variations. | 3. Neutralization in the harmonic and arithmetic averages 
altei'nately used. I 4 Disproof of them, and proof of the geometric 
mean. ^ 5. Similar tests with uneven weighting. § 6. Failure before 
such tests, extended to four or more periods, even of the method here 
advocated. | 7. Explanation ; emendation ; objection to the emendation. 
'i 8. Extent of the error in this method ; likelihood of neutralization in 
a long series. ......... pp. 324-341 



CHAPTER XI. 

THE METHOD FOR CONSTANT MASS-QUANTITIES. 

Section I. Schematization of the argument for compensation by equal sums 
of money. ^ 1. The equality properly in the sums, though it may also 
be in the prices. § 2. Schemata for mass-units equivalent at the first 
period. ? 3. Schemata for mass-units equivalent at the second period, 
and over both the periods. ...... pp. 342-348 

Section II. What this argument really proves — coincidence of the averages. 
^ 1. Same (or similar) results given by the three means (or averages) 
of price variations, with different weightings. § 2. Therefore the argu- 
ment applies to all three, each with its own proper weighting. ^ 3. 
Identification of the two averages and one mean, properly weighted, with 
Scrope's method. | 4. An argument by transposition of prices. ^. 5. 
Hence neither average better than the other or than the geometric mean 
when applicable. § 6. Consequent insufficiency in the arguments made by 
Jevons and Laspeyres. ....... pp. 348-355 

Section III. Correctness of this argument. Scrope's method. ?, 1. The 

reason for mistaking this argument as applicable only to the arithmetic 
average of price variations, 'i 2. Proof of the argument. ^ 3. The gen- 
eral principle, and the formulas of the method. ^ 4. It satisfies all the 
Propositions. ^ 5. Correctness of the geometric mean of price variations 
when applicable. ........ pp. 355-363 

Section IV. The deviation of the geometric average. § 1. An example. 
I 2. Probable direction of the deviation. ? 3. Some extraordinary 
cases. .......... pp. 363-368 

Section V. Testing in a series. I 1. In a series of three periods. ^ 2. In 
series of four or more periods. Absolute correctness of Scrope's method 
when the masses are constant. ..... pp. 368-370 



Xll CONTENTS 

CHAPTER XII. 

THE UNIVERSAL METHOD. 

Section I. Two methods suggested. ? 1. Possible extension of the two pre- 
ceding methods to all cases. § 2. Their proper forms. . pp. 371-374 

Section II. A third method, combining the two special methods. ? 1. The 
principle of the new method, and examples. ^ 2. Development of the 
method. ?, 3. Its formulae. | 4. It is a method using double weight- 
ing. Comparison with Drobisch's method. ? 5. Comparison with 
Lehr's method. ? 6. Test over three periods of these methods and of 
Nicholson's. § 7. Test over more than three periods of the new method. 
I 8. Explanation of its failure ; emendation ; unsatisfactoriness of this. 
I 9. Another test before which it fails pp. 374-396 

Section III. Comparison of the three methods. | 1. Testing of the geo- 
metric and Scrope's emended methods. | 2. Examination of the rem- 
edy, and of the test itself. ^ 3. Conditions of the coincidence of the 
three methods. ^ 4. Smallness of their divergence in ordinary cases. 
I 5. And their closeness to the truth. § 6. The choice between them : 
superiority of the emended form of Scrope's method. § 7. Remarks on 
the failure to find a perfect method. .... pp. 396-408 

Section IV. Examination of other methods, and a general test case. ^ 1. 
Testing of Scrope's method applied to the arithmetic means of the mass- 
quantities. § 2. The conditions of this method being correct. | 3. 
Need of reduction to these conditions, — likewise in Lehr's method. § 4. 
The reduction continued : the method of approach. | 5. Probable suffi- 
ciency in practice of this form of Scrope's method. §§ 6-8. A complex 
example : — the three superior methods applied to it ; other methods ap- 
plied to it ; rough methods applied to it. . . . pp. 409-433 

Section V. The missing Propositions supplied. | 1. For cases with two 
classes evenly weighted. ^ 2. For cases with constant mass-quantities. 

pp. 433-437 

CHAPTER XIII. 

THE DOCTRINE OF THE CONSERVATION OF EXCHANGE-VALUE, AND THE 
MEASUREMENT OF EXCHANGE-VALUE IN ALL THINGS. 

Section I. The doctrine described. ? 1. History of the doctrine. Need of 
the proviso in it. | 2. An empty form of the doctrine. . pp. 438-443 

Section II. The measurement, and proof of the doctrine, when the mass- 
quantities are constant. § 1. The measurement. | 2. The doctrine 
proved. ^ 3. An example, ^g 4-5. Relation between exchange-value 
in all things and exchange-value in all other things ; and a simpler form 
of it pp. 443-457 

Section III. Results when the mass-quantities vary. ? 1. The general 
measurement of the exchange-value of anything in all things. ^ 2. The 



CONTENTS Xlll 

measurement of the variation of the aggregate amount of exchange-value 
in an economic world. § 3. Approximate relation between exchange- 
value in all tilings and exchange-value in all other things. ^ 4. The 
weight of money. The commodity standard. | 5. Measurements in tliis 
standard — to find true-prices. ..... pp. 457-468 

Section IV. Need of distinguishing the two kinds of general exchange- 
value. ^ 1. An example involving a seeming inconsistency. ^ 2. Solu- 
tion of the inconsistency. ...... pp. 468-471 



CHAPTER XIV. 

THE UTILITY OF MEASURING THE VARIATIONS IN THE EXCHANGE- VALUE 

OF MONEY. 

Section I. The theoretical and practical purposes. ? 1. Necessary for in- 
vestigation into effects of such variations. ? 2. Necessary for measure- 
ments of the variations of commodities ; need of true-prices. | 3. Neces- 
sary for operating the scheme to pay debts in equal exchange-value. 
2 4. Necessary for operating tlie scheme of regulating the exchange-value 
of money. § 5. Warnings on these subjects. . . pp. 472-481 

Section II. Principles involved in the scheme of regulating the exchange- 
value of money. § 1. Eelations between variations of prices and varia- 
tions of exchange-values. § 2. The desire for everything to become 
cheaper, impossible of exchange-value. ^ 3. The desire for everything to 
become cheaper in price : nature of the standard involved, — of what kind 
of value should money be the fixed measure? ^ 4. Principles of issuance 
in order to obtain constancy of the exchange-value of money. § 5. Simi- 
lar principles of issuance if money is the standard of cost-value or of 
esteem-value. § 6. Imperfect science without measurement. 

pp. 481-495 



APPENDIX A. 

ON VARIATIONS OF AVERAGES AND AVERAGES OF VARIATIONS. 

Introduction. pp. 497-498 

Section I. On averages in general. ..... pp. 498-501 

Section II. Arithmetic averaging pp. 502-506 

Section III. Harmonic averaging pp. 506-510 

Section IV. Cases of agreement between the arithmetic and the harmonic 
averages of variations. ...... pp. 510-513 

Section V. Geometric averaging pp. 514-516 

Section VI. Comparison of the geometric average with the other two. 

pp. 516-521 

Section VII. Comparisonof averages of unequal sets. . . pp. 521-523 



XIV 



CONTENTS 



APPENDIX B. 

ON COMPENSATORY VARIATIONS 
Section I. With terms equal at the first period. 
Section II. With terms equal at the second period. 

Section III. Explanations 

Section IV. Combination of the two kinds of variations 
Section V. Transposition of terms. 



pp. 524-526 
pp. 526-527 
pp. 527-529 
pp. 529-531 
pp. 531-532 



APPENDIX C. 

REVIEW AND ANALYSIS OF THE METHODS EMPLOYING ARITHMETIC 

AVERAGING FOR MEASURING VARIATIONS IN THE 

EXCHANGE-VALUE OF MONEY. 

Introduction. pp. 533-534 

Section I. Dutot's method. p. 535 

Section II. Carli's method and its varieties. . . . pp. 535-536 

Section III. Young's method and its varieties. . . . pp. 536-539 

Section IV. Scrope's method and its varieties. . . . pp. 539-544 

Section V. Methods employing double weighting. Drobisch's. Lehr's. 
Nicholson's. Others. ....... pp. 544-552 

Bibliography. pp. 553-574 

Index. pp. 575-580 



ERRATA. 



Page 3, line 4 in notes, before Del Commercio insert Elementi 

di econmnia pubblica 

Page 3, line 8 in notes, for later read latter 

Page 9, line 2, for had read played 

Page 23, line 19, for outer read other 

Page 108, line 20, before the second insert at 

Page 136, line 2 in note, before value insert the 

Page 160, line 10, for variation read variations 

Page 198, line 6 in notes, for some read same 

P P 

Page 202, below, twice, for ^ read p2 

Page 271, in the upper equation, for 1 + 1^2^ I'^ad 1 — (3/ 

Page 317, line 3 in note, for argument read agreement 

Page 318, line 10 in note, for Sect. II. read Sect. III. 

Page 370, in the heading, for sums read masses 

Page 409, in the formula, for X2 -\- x^ read x^ + x^ 

Page 462, lines 23 and 24, for p read p^ 

Page 522, line 2, in the denominator, for Wj read n^ 

Page 530, line 26, for variation read variations 

Page 535, line 4, for substracted read subtracted 

Page 552, for V x^x^ and 1^2/i2/2 ^^^^ va^a.^ and V /^i/^a 

Page 557, line 21, for '53 read '52 



V algebra non essendo che un metodo preciso e speditissimo di ragio- 
nare sulle quantitd,, non e alia sola geometria od alle altre scienze mate- 
matiche che si possa applicare, ma si pud ad essa sottoporre tutto do 
che in qualche modo pub crescere o diminuire, tutto do che ha relazioni 
paragonahili tra di loro. Quindi anche le scienze politiche possono fino 
ad un certo segno ammetterla. 

Beccaria, 1765, 



THE. MEASUREMENT 

OF 

GENERAL EXCHANGE -VALUE. 

CHAPTER I. 
THE NATUEE OF EXCHANGE -VALUE. 



§ 1. "Value" is an ambiguous term, in the common use of 
which may be detected several meanings that deserve to be 
distinguished by special epithets. Applying the term to spe- 
cies of material things, we often have in mind their useful- 
ness or utility, as when we speak of water being very valuable 
for mankind ; and therefore what we refer to should be called 
use-value. Or we may be thinking of the particular things in 
a species, and whether the species be important or not as such, 
yet because we highly prize or esteem the particular things for 
their rarity, we attach also value to them, and this is some- 
thing which should be characterized as esteem-value. Again, 
we sometimes refer merely to the fact that the things are pro- 
duced or procured only by labor, and we value them because 
they are endeared to us by past effort and cannot be replaced 
except at the cost of more effort ; and then we should call the 
quality we are thinking of, cost-value. Lastly, we may be con- 
sidering only the fact that things once possessed may be ex- 
changed for other things, whereby a thing useless in itself to 
its owner, which he does not esteem for itself, and which per- 
haps has come to him gratuitously, may procure for him a use- 
ful and needed object, and save him the trouble of special 
1 1 



2 THE NATURE OF EXCHANGE- VALUE 

effort to produce that object, — which attribute in things is 
properly called their exchange-value. 

Having the first meaning in mind, we say a thing is more or 
is less valuable according as its class is more or is less useful as 
a whole — its " total utility " is greater or smaller — the whole 
class is less or more dispensable — our first want fiDr it is greater 
or smaller. Having in mind the second, we say that an indi- 
vidual thing, along with its mates, is more or is less valuable ac- 
cording as it is more or is less prized by people in general, which 
is generally according as their uses fov its class are greater or 
smaller and as the abundance of its class proportionally to 
their numbers is smaller or greater, or according to the magni- 
tude of what has been called its " final utility " — its usefulness 
when satisfying the last want which its class is abundant enough 
to satisfy. Thus when we say " water is more valuable than 
diamonds," we are comparing the species water with the species 
diamond ; when we say " diamonds are more valuable than 
water/' we are comparing individual diamonds with equally 
large individual drops of water. Having in mind the third 
meaning, we say a thing is more or is less valuable according 
as there is greater or smaller difficulty for people to produce it 
by finding it or by making it. Having in mind the last, we 
say a thing is more or is less valuable according as it procures 
in exchange, or purchases, more or less of other things. 

§ 2. A term rarely has several meanings without some one 
idea running through them all. In the case of "value" the 
underlying idea is that everything which we pronounce " valu- 
able " is an object of desire. Everything desired, however, is 
useful to us. But some species of physically useful things are 
so abundant that we are not aware of our desire for them until 
we stop to imagine ourselves deprived of them. Then we recog- 
nize their value — their use-value. For other things we know 
our desire because in their absence we feel the want of them. 
Thus the more useful and the more absent are they, the more 
we desire them, provided they are not so entirely absent as to 
leave us unacquainted with their utility ; and when we possess 
them, the more we prize, esteem, and value them, for fear of 



FOUR KINDS OF ECONOMIC VALUE 3 

losing them. That we desire things which cost us effort to 
produce, is shown by the fact that we spend effort to procure 
them. And anything is desirable to anybody, however useless 
in itself to him, that serves the useful purpose of providing him 
with other useful things, and so saves him the labor otherwise 
necessary to obtain them. 

By reason of this common essence, " value " may be defined 
as the valor — might, power — in things by which, in general, or 
under certain circumstances, they are rendered objects of desire. 
Things are objects of desire in four different ways. They are 
objects of desire because of their utility alone ; wherefore, 
using them, we assign to them use-value. They are objects 
of desire because of their utility and rarity ; wherefore, es- 
teeming them and holding fast to them, we assign to them es- 
teem-value. They are objects of desire because of their utility 
and the difficulty of making or replacing them ; wherefore, 
laboring to produce them, we assign to them cost-value. They 
are objects of desire because of their special utility in providing 
us with other desired objects ; wherefore, accepting or laboring 
to get even what we otherwise do not want in order by ex- 
change to get what we do want, we assign to them exchange- 
value. Apart from the common reference to desirableness, the 
four meanings of " value " are very distinct, so that they repre- 
sent four Jdnds or species of value. ^ The distinctness of these 
is also shown by the fact that the same thing, compared with 
others, may possess different degrees of use-value, of esteem- 
value, of cost-value, and of exchange-value. 

§ 3. The second and the last kind of v^alue were distin- 
guished by Turgot, who called the former" valeur estimative " 
and the latter " valeur echangeable." - The first and the last 

1 Beccariiisiiid that originally " value " meant " Iiaving force, habitude, ability 
to fultil a purpose," and this " absolute value " later became " relative and venal " , 

and meant " the power which everything has of being exchanged with all others," fji^***'^ 
^^<^Z>eZ commercio, written about 1769, ed. Custodi, Vol. I., p. 339. These two senses 
are not specifically distinct, but the latter is one species under the former as 
genus. The other species were not noticed. 

■^IVa/.eiirs et monnaies. about 1750, ed. Daire, pp. 82, 83. He also called the 
later " valeur apprcL-iative," because we generally estimate the e.xchange-value of 
things by appraising, or setting prices on, them, p. 87. 



4 THE NATURE OF EXCHANGE-VALUE 

were distinguished by Adam Smith, who called the first " value 
in use " and the last " value in exchange " or, obviously bor- 
rowing from Turgot, " exchangeable value." ^ But the latter 
term Adam Smith did not confine to the meaning of exchange- 
value. He divided " value in exchange " into two sub-species, 
which he called " nominal value " and " real value " or even 
" real exchangeable value." By the former of these he meant 
merely one instance of exchange-value, namely a thing's ex- 
change-value in one special other thing, money, — the reverse 
of its " nominal price," the quantity of money we must pay to 
purchase it ; and by the latter he meant the quantity of labor 
the thing will purchase, — the reverse of its " real price," the 
quantity of labor it costs to acquire the thing.^ Thus what is 
here called esteem-value was vaguely, and by a means of meas- 
uring it, rather than by its nature, referred to as '' real value "; 
and what is here called cost- value was given an entirely differ- 
ent appellation, " real price." ^ While Adam Smith's first and 
incomplete dichotomy of " value " has been widely accepted, 
his distinction between " real value " and " real price " has 
hardly even been noticed.^ By Ricardo the term " real value " 
was applied to cost- value, when he spoke of it as ''the quantity 
of labor and capital " — itself, in his view, a product of labor — 
" employed in producing " the tiling,'^ — in which sense he also 
indiflPerently employed the term " natural price." ® This kind 
of value he never differentiated from esteem-value, although all 

s Wealth of nations, 1776, McCulloch's ed., 1858, p. 13. 

4 See especially pp. 13, 16-17, 97, 157. 

^ The explanation of this last term is to be found on p. 14. The idea of it was 
evidently drawn from Turgot, who had said that things would have value to an 
isolated man in so far as they cost him trouble to obtain them, and had spoken of 
a commerce between man and Nature, who exacts from him labor as the price he 
must pay for what she yields, op. cit., pp. 82-83. This mere figure of speech was 
taken literally by Adam Smith. 

6 Adam Smith always used "price" in the sense of what we give (or would 
have to give) in exchange for what we acquire (or want), and "value" in the 
sense of what we «cgittre (or could acquire) in exchange for what we give (or 
have). This indeed is not a useful distinction to observe between these terms, 
and has rightly been rejected. Yet, along with the realism about labor as an ex- 
changeable commodity, it is at the bottom of many of Adam Smith's doctrines. 

'' Works, McCulloch's ed., p. 32 ; cf. also p. 171. 

« Ibid., p. 49. 



FOUR KINDS OF ECONOMIC VALUE 5 

his investigations into the causes of variations in what he called 
" relative value " hit upon the causes of variations in the rela- 
tive esteem-values of things. Thus what is here called ex- 
change-value he often referred to by the term " relative value "; 
and he also occasionally spoke of it as " nominal value," namely 
as value " in coats, hats, money, or corn," i. c, in various ob- 
jects named.^ Like Adam Smith, he does not expressly tell 
us that he is subsuming these values under " exchangeable 
value "; yet he too must do so, as he made his first division of 
" value " likewise only into " value in use " and " value in ex- 
change," '^ and it is apparent they cannot be subsumed under 
" value in use," which he identified with " riches." ^^ More- 
over there are passages in which he uses " exchangeable value " 
as he elsewhere uses " real value," ^^ while it is evident that 
" exchangeable value " must mean " relative value " when he 
defines it as " the power of purchasing [other things] pos- 
sessed by any one commodity." ^^ 

This error of confounding at times two, or even three, dis- 
tinct kinds of value together under a term applicable only to 
one of them, along with that of assigning principal importance, 
implied in the term " real," to the very one which does not de- 
serve to come under that term, has run through almost all the 
so-called " classic " political economy, notwithstanding that this 
school has constantly claimed exchange- value to be the chief 
topic it dealt with, and has rarely treated of " value " except 

* Ibid., p. 32. " Value " was sometimes used by Ricardo also in the sense of 
what is in this work called general exchange- value, pp. 293, 401 ; but he also de- 
nied this use, p. 171, and ignored it, p. 13.— As for " nominal value," this term was 
best detined by a coutemporary economist : " The nominal value of a commodity 
is strictly spealiing its value in any one commodity named ; but as the precious 
metals are on almost all occasions the commodity named, or intended to be named, 
the nominal value of a commodity, when no object is specifically referred to, is 
always understood to mean its value in exchange for the precious metals," Mal- 
thus. Principles of political economy, 2d ed., l<S3fi, p. 54. This narrow sense is the 
one above seen to have been used by Adam Smith. For it "money-value" is a 
shorter term, and free from all ambiguity. Even in its wider sense the term 
" nominal value " is not satisfactory. 

10 Works, p. 9. 

^'^ Ibid., pp. 1()9-173, against J. B. Say, who had sometimes identified " real 
value " with "value in use." 

12 //,/(?., pp. 172, 377. 

^^ Ibid., p. 49. 



6 THE NATUEE OF EXCHANGE-VALUE 

under the heading of Exchange or Distribution, and never 
under the heading of Production, which would seem to be the 
proper place for treating of cost-value, nor under the heading 
of Consumption, which would seem to be the proper place for 
treating of esteem-value. Hence it has been the cause of un- 
told amount of confusion of thought, and of wasted efiFort to get 
straightened out. Recognition of the distinction between all 
four kinds, and of the at least co5rdinate importance of the last 
kind, has not yet come into general consciousness, although a 
beginning has been made. It was not till a little over twenty 
years ago that three of the four kinds were distinguished. 
Then Jevons separated from each other use-value, esteem-value, 
and exchange- value, under the titles of " value in use," " es- 
teem, or urgency of desire," and " purchasing power, or ratio 
of exchange." ^^ It is high time that all four should be distin- 
guished, and their distinctness observed. 

It would be out of place in this work to pursue the distinc- 
tion further. Of all the four kinds, the last is the only one 
which has always been treated of by economists, and, as just 
observed in another form, many of them have asserted that 
their science is specially and even wholly concerned with ex- 
change-value. In this position may be some exaggeration, 
especially when we sever from exchange-value the other kinds 
of value which they unconsciously associated with it.'^ It is 
proper, however, that special treatises should be confined to this 
one particular kind of value, and it is the whole and sole subject 
of the present work. 

1* Theory of political economy, 2d ed., 1879, pp. 85, 87, 3d ed., 1888, pp. 78, 81. 
These terms are not in the 1st ed. published in 1871. But in the 1st edition he had 
really made the same distinctions in thought. At the same time Walras was work- 
ing out tlie laws of esteem- value in very much tlie same way, publishing them in 
the 1st edition of his Elements d^ economic politique pure, 1874. Wliile Jevons 
generally used the term "value" confined to the meaning of exchange- value, 
Walras has employed it mostly in the sense of esteem-value, occasionally using 
the term " excliangeable value " when treating of exchange-value, and frequently 
employing long and tedious phrases, from lack of short and clear-cut terms 
whereby to distinguish the kinds of value he had in mind, and sometimes fall- 
ing into confusion in consequence of dropping the long phrases. 

^5 As political economy does deal with all the kinds of value, a better technical 
name for it than Whateley's "catallactics" would be " timiotology," although 
perhaps Hearn's " plutology " is still better. 



THE EELATIVITY OF EXCHANGE-VALUE 7 

II. 

§ 1. Exchange-value is a relative quality in material things. 
A material thing has exchange- value, as it has weight, only be- 
cause of other material things to which it relates in a particular 
way. This is its exchanging for them, or its ability to exchange 
for them. Gravity is the power in a thing by which it attracts 
other things toward itself and is attracted toward other things 
by a similar power in them. Exchange-value is the power ^ in 
a thing by which it procures for its owner other things, which 
procure it for their owners by a similar power in them. As we 
cannot conceive of the gravity of one thing alone, without refer- 
ence to other things, we cannot conceive of the exchange-value of 
one thing alone, w'ithout reference to other things.^ Further- 
more, in the case of exchange-value, we know that for its ex- 
istence is required something else, namely the men who make 
exchanges.^ But it is likewise believed that for the existence 
of weight, or attraction, in material things, there is required 
some other thing as its cause, which has been variously placed. 
Now just as we can conceive of weight or attraction without 

^ Cf. Beccaria andllicardo above. McCulloeh speaks of "exchangeable value 
being the power which a commodity has of exchanging for other commodities," 
Principles of political economy, 1825, p. 213 (repeated in his ed. of the Wealth, of 
nations, p. 439). Courcelle Seneuil : " The value of a commodity is the force or 
power of exchange of this commodity," Traite d' economie politique, 1858, Vol. I., 
p. 256. And Walras speaks of a relation establishing itself between appropriated 
things such that " each one acquires, as a special property, a faculty of exchang- 
ing for each of the others in definite proportions," Elements, 1st ed., pp. 25-26; 
and on p. 48 he applies this " property " to " exchangeable value." 

- Yet Bourguin says that exchange-value is not a property, a quality, an at- 
tribute, of things — there is no intrinsic value, that is, something which we can 
conceive of in an isolated body as a quality inherent in it independently of all 
other things, adding ' ' length and weight can be so conceived in a body, apart from 
any relation, from any comparison with another thing : they are therefore intrin- 
sic qualities," B. 132, pp. 22-23; cf. p. 268. This is curious. An isolated body 
could not have weight, and we cannot conceive of its having weight, e. g., being 
unsupported, in which direction would it fall? As for length, we can conceive of 
an isolated body having length p)rovided we conceive of it having many distinct 
parts, themselves not isolated. Length is properly the distance between such 
parts, and distance is not an intrinsic quality inherent in any isolated thing. 

' The existence of the men who use, prize, and produce things, is also necessary 
for the attaching of the other kinds of " value " to things. But any material thing 
can have use-value, esteem-value, or cost-value, without reference to any other 
material thing. 



5 THE NATURE OF EXCHANGE-VALUE 

bringing into the question consideration of that other thing, so 
we can conceive of exchange-value, or of exchanges, without 
bringing into the question consideration of the men who make 
them.* At all events in our present limited inquiry it is un- 
necessary to investigate the relationship between valuable things 
and their owners, or the motives by which these are actuated in 
exchanging them in certain quantities. The owners are agents, 
and their motives are determining reasons, for the making of 
exchanges, and consequently for attaching exchange-value to 
things. In seeking to compare and measure the exchange- 
values already formed and put into things, comformably to 
qualities already under given conditions found existing in the 
things, we no more need to know the causes of exchange- value 
than we do to know the cause of gravitation. Indeed psychology 
holds somewhat the same relation to objective or formal eco- 
nomics (the study of the 'phenomena of exchanges and the laws 
of their relations) as theology to physics. We may make use 
of psychology in some branches of economics, and get back to 
causes not manifested in the phenomena themselves. This 
means that we can go further in economics than we can in 
physics. But for merely measuring the relations of exchange- 

* J. B. Clark : " The inaccuracy of the term purchasing power, often used as 
synonymous with value in exchange, consists mainly in its implying a power in 
the commodity itself to effect a purchase. Such power resides in men, not in 
things," The philosophy of ivealth, 1886, p. 88. To be sure, the power of effecting 
exchanges (the causa agendi) resides only in men ; but without possessing some 
material thing men have no power to purchase anything (although they may have 
power to produce or to earn something). More fully described, exchange-value 
is the power in things to be taken in exchange for other things (the causa fiendi) . 
It is a power in things by means of which their possessors have power of effecting 
an exchange. A derrick has no power "to effect the lifting " of anything ; yet it 
has a power by means of which men can effect the lifting of things. It would 
seem as if F. A. Walker tried to avoid Clark's objection when he defined " value " 
as " the power which an article confers upon its possessor . . . of commanding, 
in exchange for itself," other things. Political economy, 1887, pp. 5, 81. This 
would place value in the possessor, which is absurd. The same idea is expressed 
by Bourguin, who, however, immediately appended the old opinion that " value " 
resides in the commodity, saying that it is " the commodity's exchangeability for 
another commodity," B. 132, p. 20. It is interesting to note that, without think- 
ing of this question, Adam Smith described " value in exchange " as " the power 
of purchasing other goods, which the possession of that object [one having utility] 
conveys," op. cit., p. 13. This means that neither the object nor the possessor, but 
the possession has exchange-value. To conform to this position in our speech 
would only be to use much pleonasm. 



THE RELATIVITY OF EXCHANGE- VALUE 9 

value, as manifested in actual exchanges, we might be disem-^^,^ h4pu- 
bodied spirits investigating the laws of a world in which we haa 
no part and in which we could not go behind the scenes. 

§ 2. The assertion that exchange- value is a quality of mate- 
rial things, or in material things, is often denied on the ground 
that exchange-value is only an estimation which men set upon 
things and so is only in our minds — only subjective, not at all 
objective. To maintain this denial is merely to employ meta- 
physics in the wrong place. For on this line of reasoning 
there are no qualities in things, since there is no quality said to 
be in things known to us but it requires our presence for its 
existence as we know it. We are, however, permitted by meta- 
physics to speak of qualities of or in material things, on the 
ground that the material things themselves are in us in the 
same sense in which it is said the qualities we assign to them 
are in us. We may, then, be permitted to continue using the 
same popular phraseology in all cases, and to speak of the ex- 
change-value (or other values) of things so long as we speak of 
the weight, size, hardness, color, etc., of things. It is perfectly 
correct to say that exchange-value is something in our minds. 
But it is also correct, as we all really believe, to say that ex- 
change-value is something in things, and it is not correct to say 
that exchange- value is only in our minds. Exchange-value is 
not merely subjective ; it is also objective. We believe there 
is something in material things (whether these be in us or not) 
that, along with our own constitution, is the, or a, cause of our 
desiring them and behaving toward them as we do ; and it is 
this somethiug in them to which we refer when we think of 
their value, or, in particular, of their exchange-value.^ 

•"^ H. D. Macleod : " Value is not a quality of an object ... it is an affection 
of the mind. The sole origin, form, or cause of value is human desire," Theory 
of credit, 1893, Vol. I., p. 200. This omits to say what is the cause of our desire 
for things. Surely our desire for things is not wholly independent of the things 
themselves. W. L. Trenholm : "It is obvious that it cannot be any special 
quality in the thing desired which gives value to it, but that the value comes 
wholly from unsatisfied desire," The lieopWs money, \%Q'5, t^. 22Q. This ignores 
the connection between the "special quality " and the desire. — C. Menger would 
also deny the existence of exchange-value in things, afhrming that it is not a " real 
phenomenon," not only because it cannot exist in an isolated body, but because 
it is " the thought-of cause of the existing various exchange relations and of their 



10 THE NATURE OF EXCHANGE-VALUE 

This whole question, however, though important in meta- 
physics, is of no consequence in economics f for we should con- 
tinue to think of, and to investigate the relations of, exchange- 
value in precisely tlie same manner whether metaphysicians 
decided there is exchange-value in things or whether they de- 
cided it is only in our minds. The difficulties which confront us 
in the metrology of exchange- value do not arise from its sub- 
jectivity, if it be only subjective, and would exist the same if it 
be also objective. They arise solely from its extreme variabil- 
ity, whatever be the cause of this. But it has been necessary 
to point this out here, on account of the frequency with which 
the opinion is advanced that we can find no invariable standard 
of exchange-value because of its being only subjective unlike 
weight and other qualities which we have succeeded in measur- 
ing with tolerable exactness.^ 

§ 3. A thing which has exchange-value has exchange-value 
in relation to other particular things, or, by combining these 
under the same terms, to other particular kinds of things. 
Individuals of one kind exchanging in certain proportions for 
individuals of this or of that or of any other kind, it is said 
the former kind has a certain exchange-value in this kind, in 

variations, which cause is in our tlioughts ascribed to the commodity in ques- 
tion," art. Geld in the Handworterbuch der Staatswissenschaften, Jena, Vol. 
III., 1892, p. 740. But at the same time he allows it to be an Austauschmoglich- 
/keii belonging to things. Tlien why not also aTauschkraftf And if we define 
exchange-value as such, it would belong to things. — The denial is made also on 
the ground that exchange-value is only a relation. So Denis, B. 100, p. 171. 
The error of this will be seen presently. 

s Of course to complain, with H. Dabos, La theorie de la valeur, Journal des 
Economistes, March, 1888, p. 406, that exchange-value is not a physical property 
of things, like their color, density, porosity, etc., is to go out of one's way in 
search of trouble, siuee economists should leave physical properties to physicists 
and concern themselves with the economic properties of things. 

'^. g., J. P. Smith, B. 7, p. 39.— T. Martello says that speaking of a fixed 
unit of value is like speaking of a fixed unit of love, La moneta e gli errori che 
corrono intorno ad essa, 1881, p. 401. — It is only a step to say that, because value 
is only in us, there is no such thing as the value of a material object. This step 
has been taken even by Walras in the very work in which he started out by de- 
scribing exchange-value as a property acquired by things under certain circum- 
stances ; whence he concluded that the term " franc " cannot refer to the value 
of a piece of metal, there being none, but only to the piece of metal itself. 
Elements, p. 147 (2d ed., p. 172). But this has not prevented him from later 
writing a work on a method of regulating variations in the " value of money," 
B. 69. A similar position is adopted by Bourguin, B. 132, p. 38. 



THE RELATIVITY OF EXCHANGE-VALUE 11 

that kind, and so on, — a contracted expression for " in relation to 
the other kind." The excliange-vahie of one thing in another, 
or of one kind of thing in another kind of thing, may be called 
a pai'tieular exchange-value of the thing, or of the kind of thing ; 
so that we may speak either of the particular exchange-value 
of one thing in another thing or of the particular exchange- 
value of one kind of thing in another kind of tiling.'^ 

Such a particular exchange- value of one thing in another, or 
of one kind in another, is not the other thing itself, or the 
definite quantity of the other kind of thing, procurable in ex- 
change, as has often been carelessly said.'' Nor is it simply a 
relation, or ratio, between the two things.'" Indeed it is diffi- 
cult to see what meaning there is in such a definition of ex- 
change-value. If a bushel of wheat exchanges for two bushels 
of barley, is the exchange-value of wheat the ratio two f We 
commonly say in this case that the exchange- value of wheat is 
twice that of barley ; but this is only a statement of comparison 
between the exchange- values of equal quantities of wheat and 
of barley. It does not pretend to tell us what the nature is of 
the exchange-value of either of these things." It is plain that, 

* This is anotlier term for " nominal value " as above defined by Malthiis. 

' E. g., Adam Smith, as already observed in Note 6 in Sect. I.; also J. B. Say, 
Traite d' econoinie politique, 5th ed., 1826, Vol. II., p. 156 ; J. S. Mill, Principles 
of political economy, 1S48, ed. 1878, Vol. I., p. 588. Macleod, op. cit., Vol. I., 
pp. 112-113, 170, 172. — The error has been pointed out by .levons, op. cit., 1st ed., 
pp. 81-82, 3d ed., p. 78. 

^° E. g., "Value in exchange is the relation of one oljject to some other or 
others in exchange," Malthus, op. cit., p. 50, and similarly again, p. 61. — Beccaria 
had said: " Value indicates the proportion of one quantity with another," op. 
cit., Vol. II., p. 8. There is a gi-eat difference between these expressions. 

1^ Thus Jevons himself was in error when he added : " Value in exchange is 
nothing but a ratio, and the term should not be used in any other sense," loc. cit. 
All that he had a right to say was that the ideas of the other kinds of value should 
be excluded from the idea of this kind of value. He had himself previously de- 
clared of this kind of value: "Value is a vague expression for potency in purchasing 
other commodities," B. 22, p. 20 ; and he continued to use the term in this sense, 
of. Investigations, p. 358. What absurdity this position may lead to is well illus- 
trated in the following: "Value ... is a relation between certain things to 
which men attribute value," Trenholm, op. cit., p. 246. — Walras has gone to a 
peculiar extreme also here. He says: " Strictly speaking, there are no values, 
there are only relations of values," Elements, 1st ed., p. 189 (again in B. 71, p. 4). 
(Perhaps the meaning intended to be conveyed in this sentence is that exchange- 
value is a relation between esteem-values.) — Carelessness in trebly describing 
value as (1) ?k ratio, (2) the quantity of another or other things, (3) purchasing 



12 THE NATURE OF EXCHANGE-VALUE 

as already shown of exchange-value in general, the exchange- 
value of one thing in another is the power in the one thing of 
exchanging for the other — a power which necessarily presup- 
poses certain relations of quantity between the things exchanged, 
but is itself very different from those relations. Thus the ex- 
change-value of one thing in another is neither the other thing 
nor the relation between the two ; but it is the power in the one 
which can exist only in connection with a similar power in the 
other, and can be estimated, as we shall see, by the relation be- 
tween the quantities exchanged. 

§ 4. A thing has many such particular exchange-values — as 
many, in fact, as there are kinds of things with which it 
can exchange. Now several, many, or all of these particular 
exchange- values, as they exist together, may be combined into 
a single concept, and so provide us with the idea of the thing's 
exchange-value simply so called. ^^ This idea of simple ex- 
change-value, like that of the gravity of the heavenly bodies, is 
difficult to grasp at first. Yet it is a necessary idea, which we 
all do inevitably form with various degrees of definiteness and 
accuracy. ^^ It is at any given time and place a single exchange- 

poioer, is also shown by J. L. Laughlin, Facts about money, Chicago, 1895, pp. 75- 
76, 147, 192, and Parsons, B. 136, pp. 81-82. 

^" J. L. Shadwell : " Tlie human mind can only compare two things at once, 
and when it is said that a commodity has a certain power of purchasing all other 
commodities, the words, though they may be pronounced, written, and printed, 
do not really present any idea to the mind. The power of gold to purchase silver 
is a definite idea, and so is its power to purchase copper ; but the power of gold 
to purchase silver and copper means nothing at all," System of political economy, 
London, 1877, p. 93. The only reason offered why we cannot strike an average is 
that " we have no standard by which to measui'e the objective importance of dif- 
ferent articles," p. 95. This is a difSculty, to be sure, but one by no means in- 
superable, — and one which has long been discussed, it forming one of the definite 
problems in our subject (here to be treated in Chapter IV.). Few persons would 
admit in their own case the impotency here claimed for every human mind. 

^ 3 Bourguin asserts that a thing has no " value in general," but only "particular 
values" (by "value " always meaning " exchange-value," B. 132, p. 3). His only 
reason seems to be, because exchange-value is not a property inherent in things, 
p. 22. Cf. p. 135, where he says : " The [purchasing] power of money is only a 
word ; it designates, not a quality, an intrinsic value, but an ensemble of rela- 
tions which have nothing in common, not being equations between magnitudes of 
the same kind." But if the particular exchange-values are not intrinsic, neither 
would the general exchange-value be intrinsic, and no reason is shown why there 
is no non-intrinsic general exchange-value except the statement that the relations 
{i. e., the particular exchange-values) have nothing in common. But this is not so, 



THE RELATIVITY OF EXCHANGE-VALUE 13 

value somehow made up of many particular exchange-values, 
A thing has many exchange-values in other things separately ; 
it has one exchange-value in other things collectively. Or 
rather the more correct statement is that when we do reach this 
idea of a thing's simple exchange- value we must view it as the 
thing's only real exchange-value/* and regard the particular 
exchange-values as merely this one and the same exchange- 
value in its various relations to the similarly single and simple 
exchange- values of other particular things. One of the tasks 
of political economy is to explicate and render more intelligible 
this conception of a thing's exchange-value in other things, or 
in all other things. To contribute to the accomplishment of 
this task is one of the objects of these pages. It may be pre- 
mised that we shall find an exchange-value of a thing in all 
things, including itself, — an idea never yet distinguished from 
that of the exchange-value of a thing in all other things. These 
two ideas have some points of contact, and both may lay claim 
to the title of a thing's exchange-value simply, or its general ex- 
change-value}^ It is plain that we have no right to speak 
simply of a thing's exchange- value, if we have in mind only its 
exchange-value in some other thing, or in a few other things. 
Speaking of its exchange-value simply, we should be using 
language wrongly unless we referred to its exchange- value in 
all, or in all other things — or at least in all others to which we 
practically can refer, — that is, to its one simple exchange- value 
as measured by comparison with the simple exchange-values of 
all the other things. "" 

as the particular excliange-values of a thing have in common powers of exchanging 
for certain quantities of other things, and these powers are magnitudes of tlie same 
kind. As well say the attraction of the earth for the moon is different in kind 
from the attraction of the earth for the sun. Bourguin elsewhere says he will use 
the term "the value of a thing," in the singular, meaning the ensemble of all its 
particular exchange-values, only for convenience, p. 23 (and he will attempt to 
measure variations in this, ensemble, pp. 138-139). But we may be sure that when 
a term is found to be convenient, it expresses an idea or concept. 

1* But of course not its only " real value." 

^^ This term was much used by J. S. Mill. In the same sense " general value " 
was used by Hallam, View of the state of Europe in the 3fiddle Ages, 181G, Chapt. 
IX., Part II. 

^^ Naturally in speaking of exchange-value simply we do not mean something 
without relation to any other things. Neglect of this has led Macleod into curious 



14 THE NATURE OF EXCHANGE-VALUE 



III. 

§ 1. Exchange-value is quantitative. A certain quantity of 
one thing exchanges for a certain quantity of another thing. If 
it exchanges for more of the other, its exchange-value in the 
other is proportionally greater. If it exchanges for less of the 
other, its exchange-value in the other is proportionally smaller. 
If it exchanges for the same quantity of the other, its exchange- 
value in the other is the same. Thus the exchange-value of 
one thing in another may have different degrees of intensity. 
This being so of its exchange-value in any one other, it is so of 
its exchange-value in every other, and consequently of its ex- 
change-value in all other things, or of its general exchange- 
value. And so with the exchange- values of everything. 

Such variations in exchanges, and consequently in exchange- 
values, be it said in passing, may occur at different times and 
places. Consequently to speak of a thing's exchange-value in 
another thing, or in general, is to refer to its exchange-value at 
a given time and place. Exchange- value is something temporal 
and local. In omitting explicit declarations, we generally im- 
ply that we are dealing with exchange- value at the same place 
and are referring to changes happening in time. 

§ 2. From the above comparisons it results that, the quantity 
of one thing being assumed, its exchange-value in another is pro- 
portional to the quantity of the other it exchanges for (Proposition 
I.). Consequently we can measure the exchange- value of one 
thing in another at two dates (or places) by the relative quan- 
tities of the other it exchanges for at the two dates (or places). 

error. He says : What is wanted by economists who seek an invariable standard 
of value is " something by which they can at once decide whether gold is of more 
value in A. D. 30, in A. D. 1588, or in A. D. 1893 ; in Italy, in England, or in 
China; without reference to anything else," Theory of credit. Vol. I., p. 212. 
And so he had long before said : " As no single body can be a standard of distance 
or equality [without reference to others] , so no single object can possibly be a 
standard of value" without reference to others. Theory and practice of banking, 
1875, Vol. I., p. 16, and similarly again, p. 77. Of course the exchange-value of 
a single body ivith reference to others may be a standard of exchange-value, just 
as the length of a single body — a certain distance between its extremities — com- 
pared with others, may be a standard of length. (But some of the economists 
referred to treated of cost- value, and so took account only of cost of production. ) 



THE QUANTITATIVENESS OF EXCHANGE- VALUE 15 

These quantities of the other, we must remember, do not con- 
stitute the exchange-value of the one in that other. The ex- 
change-value of the one in the other is its power of acquiring 
that other for its owner. The magnitude of this power over 
the other is manifested by the effect it accomplishes in exchang- 
ing for that other, that is, by the quantity of that other tiling it 
acquires.^ Here we are treating merely of the power of the one 
thing over the other, or its exchange-value in the other, and 
therefore need pay no attention to the power of the other, or to 
its exchange-value. But if we were comparing the exchange- 
value of the one with the exchange-value of the other, or trying 
to measure how much exchange-value the former is manifesting 
when it is exchanging for the latter — by " exchange-value" here 
meaning exchange-value simply, or general exchange- value, — we 
should have to take into consideration also the general exchange- 
value of the latter. 

It is plain that at any one place individual things exactly alike 
physically will always exchange for one another indifferently, or 
have the same exchange-value in their own class; and also will 
exchange for the same quantities of other things, or have the same 
exchange-value in other things: that is, all the individuals in a 
homogeneous class have the same exchange-value (Proposition II.). 
Like things are not generally exchanged, because there is gen- 
erally no object in making such an exchange. But such an ex- 
change is possible, and sometimes occurs. Dealers who store 
wheat together, probably get back different wheat, and so have 
exchanged wheat for wheat. Also in the case of money, we fre- 
quently exchange, say, ten dollars in one piece for ten dollars 
in two or more pieces. It is evident that when such exchanges 
are made people do not exchange more for less of the same 
thing. Now a bushel of wheat and any other bushel of wheat 
having the power of exchanging for a bushel of wheat, every 
bushel of wheat has the same exchange- value in wheat (all of 
the same quality). And when one kind of things is exchanged 
for another kind, it is indifferent which of many like individuals 
is given or received. Therefore, each of these in the one class 

^Cf. Courcelle Seneuil, op. cit., Vol. I., p. 243. 



16 THE NATURE OF EXCHANGE- VALUE 

having the power of exchanging for the same thing in the other, 
they all have the same exchange-value in that other kind of 
thing. And for the same reason they have the same exchange- 
value in any and every other kind of things, consequently in 
all other kinds of things together, that is, in exchange-value 
simply, or in general exchange-value. It may be that the dif- 
ferent individuals which we include under a class with the same 
name are generally not exactly alike, and perhaps they never 
are exactly alike — it has been maintained that no two grains of 
wheat are ever exactly alike ; — but they are often nearly enough 
alike to pass in practice for alike. Of course what is here said 
refers to materials in the same form only ; for the form itself has 
utility. A cubic foot of wood in a lumber yard has not the same 
exchange-value as a similar cubic foot of w^ood in a building. 
It is plain also that, the exchange-value of a quantity of one 
thing in another being given, the power of acquiring quantities of 
the latter by means of the former is proportional to the quantity of 
the former employed in exchanging for the latter (Proposition 
III.). For example, if one bushel of wheat has the power of 
exchanging for two bushels of barley, two bushels of wheat 
have double this power, that is, they have the power of ex- 
changing for four bushels of barley ; for, according to the pre- 
ceding proposition, the second bushel of wheat has the same 
power as the first. This proposition does not mean that if ten 
bushels of wheat have the power of exchanging for a diamond 
of a certain size, twenty bushels of wheat have the power of 
exchanging for a diamond of twice that size. It means only 
that they have the power of exchanging for two such diamonds- 
It also does not mean that if one bushel of wheat actually ex- 
changes for two bushels of barley, any quantity of wheat might 
have been exchanged for t^dce the same quantity of barley ; 
for this would require that the exchange-values should remain 
the same whatever be the quantities offered in the market. It 
means only that such proportional exchanges can be made while 
the exchange-values do remain the same. 

§ 3. Again, of two kinds of things the quantities which ex- 
change for each other are equivalent in the sense that the exchange- 



THE QUAXTITATIVENESS OF EXCHANGE- VALUE 17 

value of the one in the other is equal to the exchange-value 
of the latter in the former (Proposition IV.). This looks 
as if it were an expletive proposition, like those which have 
preceded, springing from the meaning of the terms employed, 
or like saying that the distance of the snn from the earth is 
eqnal to the distance of the earth from the sun. Yet it may 
be objected that, the power of wheat to acquire barley being 
measured by the quantity of the barley, and the power of 
barley to acquire wheat being measured by the quantity of the 
wheat, as the quantity of barley and the quantity of wheat are 
two distinct and generally unequal things, the equality of the 
two exchange-values is not shown by a comparison of these 
quantities. If a proof be demanded, however, a proof is forth- 
coming. We have seen that a bushel of wheat has the power 
of acquiring a bushel of wheat. But, according to our suppo- 
sition, two bushels of barley have the power of acquiring a 
bushel of wheat. Therefore two bushels of barley have the 
same exchange-value in wheat as one bushel of wheat. And 
similarly a bushel of wheat has the same exchange- value in 
barley as two bushels of barley. Therefore, having the same 
exchange-value both in barley and in wheat, a bushel of wheat 
has the same exchange-value in barley as the two bushels of 
barley have in wheat.^ 

To say that the quantities of things which exchange for each 
other are equivalent in the sense of having the same exchange- 
value simply, that is, the same general exchange-value, needs 
still further proof. 

Now, of two kinds of things the quantities which exchange for 
each other exchange for the same quantity of any other kind of 
things, and therefore have the same exchange-value in that other 
kind of things (Proposition V.). The first part of this proposi- 
tion is the law of stable equilibrium in an open and free market, 
which equilibrium must exist on the average in the long run. 
It is possible for fluctuations to occur at times, but then forces 
are set at work to restore the equilibrium. For instance, while 

2 The case is comparable with weights. We should never know by the balanc- 
ing of two bodies in opposite scales that they are of equal weight except by 
alternating them, or by employing " double weighing." 
2 



18 THE NATURE OF EXCHANGE-VATiUE 

one bushel of wheat exchanges for two bushels of barley and 
for three bushels of oats, if it should happen that some one is 
willing to give up four bushels of oats for two bushels of bar- 
ley, immediately those who have wheat and want oats would 
exchange their wheat for barley and it for this man's oats, and 
those who have barley and want wheat would exchange their 
barley for this man's oats and them for wheat. This man's sup- 
ply of oats would then be soon exhausted, and he would retire 
from the market. The condition which renders useless such 
roundabout exchanges (which in money exchanges between dif- 
ferent places are called arbitrages) — a condition which those 
roundabout exchanges themselves tend to produce, — exists in our 
example when three bushels of oats exchange for two bushels 
of barley. Then, representing one bushel of 'each by A, B and 
C, and equivalence by the sign ^=^, we have between the kinds 

^-^ 2 B 

^ 3 C 

of things the interrelation here depicted. That any two of 
these quantities have the same exchange-value in the third is 
apparent. 

This being so of any two things in any third, it must be true 
that the quantities of everything which exchange for each other 
have the same exchange-value in all things beside themselves. 
But as their own exchange-values are equal each in the other, 
these may be added, and we have : Of everything the quantities 
lohich exchange for each other have the same exchange-value in all 
other things and in all things, that is, the same general exchange- 
value, or the same exchange-value simply (Proposition VI.) ; which 
is what we wished to prove. 

It follows also that all the many particular exchange-values of 
one thing in other things singly are singly equal to the thing's gen- 
eral exchange-value, and to one another (Proposition VII.). The 
exchange-value of wheat in barley is, for instance, its power of 
acquiring two bushels of barley, and its exchange-value in oats, 



THE QUANTITATIVENESS OF EXCHANGE-VALUE 19 

its power of acquiring three bushels of oats ; but the two bushels 
of barley and the three bushels of oats have the same general ex- 
change-value, therefore the one bushel of wheat is acquiring the 
same exchange-value, to which its own is equal, whether it be 
exchanged for two bushels of barley or for three bushels of oats. 
It is evident, moreover, that when a bushel of wheat is used to 
acquire barley, it is manifesting its whole exchange- value, but 
only in relation to barley ; and when it is exchanged for oats, it 
is manifesting its whole exchange-value, but only in relation to 
oats. Therefore it is manifesting the same exchange-value in 
both cases, but in different relations. The two particular ex- 
change-values, as we have already noticed, are the same as the 
general exchange-value, are equal to it, and consequently are 
equal to each other. And so of all the particular exchange- 
values of any one thing. 

§ 4. To measure even the particular exchange-value of one 
thing in another is not an easy task ; for at the same place 
during the same period of time the same quantity of the same 
kind of thing may be exchanged for various quantities of an- 
other kind of thing. It is sometimes said that the exchange- 
value of one thing in another is determined by an actual ex- 
change. This is not so, as it often happens that in an actual 
exchange the one party rejoices over a good bargain and the 
other is worried lest he have made a bad bargain — /. e., the one 
thinks he has got more, the other less, than the thing given 
was worth. We also ascribe exchange-value to things never 
exchanged, and especially we want to know the proper ex- 
change-value of a thing which we are going to part with, before 
we effect its exchange for anything else. AVe estimate ex- 
change-values rather by the general run of exchanges of simi- 
lar other things. The particular exchange- value of one kind 
of thing in another kind of thing is not an affair of a single 
exchange, but of many. The single exchanges may fluctuate 
around an average, which is what we call the exchange-value 
of the thing (the class) during the period in question. When 
certain kinds of things are habitually exchanged in large quan- 
tities, the fluctuations in short intervals of time are not apt to 



20 THE NATURE OF EXCHANGE- VALUE 

be large, so that in these cases it is tolerably easy to determine 
their particular exchange- values within narrow limits of error. 

§ 5. If the particular exchange-value of one thing in another 
is sometimes difficult to estimate, it is always difficult to esti- 
mate its general exchange-value • for this includes not only the 
determination of its many particular exchange- values, some of 
which are sure to be troublesome, but also the combination of 
these into a whole. This problem, however, may perhaps ad- 
mit of a satisfactory theoretical solution, whereupon the diffi- 
culties will reside only in the practical details. 

The labor of finding the particular exchange-values of things 
would be almost infinite if we were to undertake to find all the 
particular exchange-values of all things. For as one thing has 
a particular exchange-value in another thing, this has a par- 
ticular exchange-value in it, so that there are two relations of 
particular exchange- value between every two things ; and, 
therefore, among a hundred kinds of things, each one of the 
hundred having a particular exchange- value in every one of the 
other ninety and nine, there would be 100 x 99 =9,900 par- 
ticular exchange-value relations, and among two hundred there 
would be 200 x 199 = 39,800, and so on in rapidly increasing 
progression. Through this maze of interminable interrelations 
we have been enabled to make our way, as is well known, by the 
invention of money, which, as is said, serves as the " common 
denominator " for all other things. By means of prices, money 
acts as a perfectly satisfactory measure of the exchange-values 
of other things in the same place at the same time. Now money 
alone is brought into direct relationship with every other ex- 
changeable thing, the relations between these others being inter- 
mediated through their relations to money. Hereby our atten- 
tion is confined to the particular exchange-values of all other 
things in money, as indicated by their prices, or, which is the 
same thing inverted, to the particular exchange-values of money 
in all other things singly. Consequently it is primarily only 
the general exchange-value of money in all other things col- 
lectively that we are concerned with measuring ; for after 
measuring it and finding its constancy or variation at different 



THE QUANTITATIVENESS OF EXCHANGE-VALUE 21 

times, or in different places, we can measure the constancy or 
variation of any other thing in general exchange-value by its 
known constancy or variation in relation to money. It is not 
an uncommon opinion that it is easier to estimate the variation 
in exchange-value, simply spoken, of any commodity than it is 
to estimate the variation in the exchange-value of money, on 
the ground that in the former case we only have to notice the 
variation of its price, while in the latter we have to consider 
variations of all prices. But the variation of a commodity's 
price only tells the variation of the commodity's particular 
exchange-value in money, and gives us no information concern- 
ing the variation or constancy of the commodity's exchange- 
value simply so called, or general exchange-value, until we 
know the variation or constancy of the general exchange-value 
of money. Thus the former calculation really presupposes the 
latter. And as a matter of fact, not only is it easier to meas- 
ure the constancy or variation of the general exchange-value of 
money than of anything else, but money is the only thing of 
which the general exchange-value can be measured by us di- 
rectly ; for we should never be able to find all the particular 
exchange-values of any other thing without taking account of 
its and of the other things' prices, or exchange-values in 

money. , 

§ 6. Because exchange-value is quantitative, to conceive ot 
exchange-value involves measurement. The measurement may 
be rough or exact, but measurement there must be. And we 
make measurement not only of particular exchange-values, but 
also of general exchange-value— especially of money's general 
exchange-value. Everybody has some notion of " money's 
worth," and some opinion as to whether through a course of 
years this worth or exchange-value has remained constant or 
varied, or whether it is greater in one place than in another. 
Only this notion is generally very badly formed and left vague 
and unprecise, wherefore the opinion is generally weak and ir- 
resolute, or inclines in favor of constancy merely through lack 
of proof of variation. Political economy, if it be a science, 
cannot avoid the duty of attempting to rid this notion of its 



22 THE NATURE OF EXCHANGE- VALUE 

vagueness ^ and to provide a method of measuring the general 
exchange- value of money with theoretical precision as a model 
to be realized as closely as possible in practice/ 

ly. 

§ 1, In measuring quantities we must bear in mind that we 
are not concerned with the causes of their constancy or of their 
variations. In measuring from year to year the weight and 
tallness of a boy, we have nothing to do with the causes that 
make him grow. To measure variations, and to explain them 
by pointing out their causes, are two distinct operations.^ And 
the former is the primary ; for we can be scientifically prepared 
for investigating the cause of variations only after measuring the 
variations with scientific precision. When direct measurement 
of things we desire to measure is not feasible, we frequently 
measure them, as less apparent causes, by their more apparent 
effects, if we can eliminate all other causes, as in the familiar 
example of heat, by expansion. But less apparent effects we 
rarely attempt to measure by their apparent causes, since these 
effects generally escape our control and we cannot know whether 
they are operated upon only by the causes employed to measure 
them by. And equally apparent effects it would be useless to 
measure by their equally apparent causes, as we can measure 
them directly. To attempt, then, to measure apparent effects 
by their less apparent causes, would be the height of folly. 
Now the variations of exchange- values are more apparent than 

^ J. S. Mill spoke of "the necessary iudefiniteness of the idea of general ex- 
change-value," op. cit., Vol. II., p. 102. Of course until it is made definite, 
this idea will remain indefinite. But what shows the necessity of its remaining 
indefinite ? And why should the task of clarifying it be shirked by any econo- 
mist? 

* J. B. Say said we cannot "measure" exchange-value at diflTerent times and 
places, but we can only " appraise" it, or form " approximative valuations," be- 
cause of the absence of an invariable measux-e — i. e., because we cannot make an 
absolutely exact measurement, op. cit., Vol. II., pp. 85, 93. This distinction is 
too hard and fast, since an inexact measui-ement is a measurement, and if we 
took his statement literally, we should have no measures, as none is absolutely 
invariable. We ought at least to aim at more than appraising ; we ought to aim 
at measuring. 

1 Cf. Jevons, B. 22, pp. 21, 59 ; Nicholson, B. 94, pp. 304-305. 



fi.V 



MENSURATION NOT CONCERNED WITH CAUSES 23 

their causes ; for, difficult though it be to measure accurately a 
variation in the general exchange-value of money, it is less 
difficult than to assign all the exact causes which have produced 
it (although it may be easier to adduce many possible causes in 
a general way than actually to measure the precise variation). 
Therefore it is especially absurd in .political economy to say 
that in attempting to measure exchange-value we must pay at- 
tention to its causes. On the contrary, we should be especially 
careful to drop all consideration of its causes. 

§ 2. It is necessary to state this obvious truth because of the 
prevale^e of the opposite opinion in this one subject alone 
among all subjects of metrology. Thus, for example, a writer 
has recently asserted : " To measure the variations of money 
there is required not only detailed knowledge of prices, but 
also of the causes which produce the variations."^ And 
another has said that this measurement is impossible because 
of the impossibility of knowing all these causes.^ It would be 
difficult to match such assertions with similar assertions in any 
^^branch of science.^ Yet opinions of this sort in political 
^omy have no less an authority than that of Ricardo.^ 

The reason for this error is twofold. It lies in the confusion 
between cost-value and exchange-value, and in the doctrine that 
" value," including exchange-value, is determined by the labor- 
cost of production. But it is only cost-value which is propor- 
tionate to labor-cost; and even if exchange-values were de- 
termined by the relative labor-costs, it would not follow that 
the exchange-value of a thing can be measured by its labor- 
cost, but onjy by its labor-cost compared with the labor-costs 
of other tilings,^ which comparison would be more difficult 
2 Nitti La mimra delle variaziom. di valore dclla moneta, 1895 (Quoted from 
^lui, j^an^t^iA, „ 9fiO ^ Cf V Pareto, Cours d'&conorme poh- 

the Economic Journal, Vol. v., p. ^ou.j «^r. v. xdici, , 

tiqite, Lausanne 1806, Vol. I., p. 266. 

: "^TZ::^, I "Whewell spea.s of " an ^^^^^l^^:: :^X. 
tive science, that we must first obtain the measure and ascertain «^«/;"^^«; j;!' 
nomena, before we endeavor to discover their causes,^ Phrlosophy of the .nducHve 
sciences, 1847, Vol. II., p. 240. 

erw^^a^'e'l^xrly Pointed out by E. Torrens, Essay on tke produciton 
0/ W;rLXflt2i:p'p. .., 56, and by S. ^f^^^^^^^'Z 
the nature, measure, and causes of value, l^onAon 182o, pp. 6 U, 17 K, 
Letter to apolitical economist, 1826, pp. 53-54. 



24 THE NATURE OF EXCHANGE-VALUE 

than the comparison between their actual ratios of exchange, 
and so would be worthless even if that doctrine were univer- 
sally true, and is especially worthless since that doctrine is not 
universally true (and, in fact, never even pretended to be), 

§ 3. Another objection is a variation upon the same theme. 
This is that the measurement of the general exchange-value 
of money, however successfully made, is useless, because it 
gives no information about the causes of the variations/ It 
may seem strange that any economists could raise such an objec- 
tion ; for in no subject does a measurement disclose causes, and 
in nothing else is mensuration reproached for its inability to do 
so. The explanation is that what these economists want is 
really a measurement of cost-value, and so they are dissatisfied 
with what turns out to be a measurement of exchange-value 
alone, which being called simply a measurement of " value," 
may have seemed to them to give promise of being a measure- 
ment also of cost-value.'^ What they want is a measurement 
of the variations in the cost-values of commodities and in 
the cost-value of money (gold) — or rather, a measurement of 
the variations in money as a standard of cost- value. Or even 
sometimes they want such a measurement of money as a stand- 
ard of esteem-value ; for some of them recognize that gold is 
a semi-monopolized product, with " value " enhanced by rarity 

■^ So D. A. Wells, Recent economic changes, 1889, p. 121. Cf. Malthus, op. cit., 
p. 120, and McCulIoch, Political economy, p. 214, (Note to Wealth of nations, 
pp. 439-440). 

^ Malthus ofiFers a peculiai* example of confusion. He wrote : " The exchange- 
able value of a commodity can only be proportioned to its general power of 
purchasing [general exchange-value] so long as the commodities with which it 
is exchanged continue to be obtained with the same facility," op. cit., pp. 58-59. 
Thus although he expressly uses the term " exchangeable value," he distinguishes 
it from purchasing power or exchange-value proper, and identifies it with cost- 
value. He does so still more plainly when he amplifies the term into "intrinsic 
value in exchange," p. 60. Yet the idea of exchange-value always attaches to 
these terms, being embodied in them. The fundamental fault, which runs through 
all his long disquisition on the measure of value, and prevents it from reaching a sat- 
isfactory conclusion, is the fact tliat he is seeking the impossible — ^a single measure 
both of exchange- value and of cost-value. — Eicardo had the same idea as Malthus 
when he wrote: " Why should . . . all commodities together be the standard, 
when such a standard is itself subject to fluctuations in value ? Works, p. 16(3, i. e., 
in cost- value (and in esteem-value). But of course their fluctuation in other 
kinds of value is not a reason why they should not be the standard of exchange- 
value if they be found not to fluctuate in this. 



MENSURATION NOT CONCERNED WITH CAUSES 25 

above its labor-cost — that is, with esteem-value greater than its 
cost-value. And this measurement they think can be made by 
examining the variations in the costs of production of commo- 
dities and allowing for them in the variations of prices ; for 
they consider that the causes of the variations of prices lie to 
a great extent in the altered costs of production of the com- 
modities themselves, leaving at times a remainder, which is the 
sole cause of the variation of prices due to the influence of 
money, and consequently the only measure of the " inner " 
variation of the " value " of money, or of its appreciation or 
depreciation f — all which may be correct enough, but only of 
the cost-value of money, or of its esteem-value, and not of its 
exchange-value/" 

Now these economists have a perfect right to their wish, and 
if they succeeded in inventing a method by which the constancy 
or variation of the monetary standard in cost-value or in esteem- 
value could be measured, they would be rendering an important 
service in political economy. But there is room in political 
economy for the measurement of exchange-value as well as for 
the measurement of cost- value or of esteem-value. If political 
economy be at all a science of exchanges, its need for a meas- 
urement of exchange-value is very great, and undiminished by 
any further want which may be felt for a measurement of other 
kinds of value. Let us then pursue our course of seeking the 
method of measuring general exchange-value, undeterred by 
the fact that this measurement is not a measurement of some- 
thing else. 

® Cf. W. Lexis In the Verhandhingen der deutschen Silbcrkommission, 1894, 
p. 153 ; K. HelfFerich Die Wilhrungsfrage, 1895, p. IS. Probably this was the 
opinion also of Eicardo in the passage referred to in Note 5. 

^^ They even then object to the measurement of exchange-value as being mis- 
leading (so especially G. JI. Fiamingo, The measure of the value of money accord- 
ing to European economists, Journal of Economics, Chicago, Dec. 1898, pp. 74- 
75) for a reason whicli amounts to tliis : that people may mistake a conclusion 
concerning exchange-value for a conclusion concerning cost-value or esteem-value. 
But the best May for them to keep others from sucli error is for themselves to 
avoid the confusion of tliought between the several kinds of value. 



CHAPTER II. 

THE COEEELATION OF EXCHANGE-VALUES. 

I. 

§ 1. The mutual exchange-values of two things, each in the 
other, are subject to a very simple law. If A rises in ex- 
change-value in B, ipso facto B falls in exchange-value in A ; 
and reversely. A variation of one thing's exchange-value in an- 
other is an opposite variation of the latter' s exchange-value in the 
former (Proposition VIII.). Of two boys at one time evenly 
tall, if the one grows taller than the other, the other ipso facto 
becomes shorter than the first. Here we are apt to say the 
second boy has not grown shorter, but may even have grown 
taller, only the first has grown more. We do so because we 
have other things in mind, with the lengths of which we com- 
pare these lengths as well as with each other. If the boys 
were in space alone and it were impossible to compare their 
lengths with any other lengths, a change in their relative 
lengths would enable them only to say, " You have become 
taller than I " and " You have become shorter than I." 

It may be said that if we knew in which of* the boys the 
cause of the change lay, we should then know which of the 
boys has changed and which not (or how much each) — in 
something called absolute length. But it is evident that we 
cannot seek for the cause of a change until we know the change, 
and so we must know which of the boys has changed before we 
can know where the cause lies. Similarly with the case of ex- 
change-value. Compared with the exchange-values of other 
things, and with a view to the variations of their exchange- 
values in other things, the change between A and B may be a 
change only in the one, or even both may have changed in the 

26 



BETWEEN TWO THINGS 27 

same direction, but the one more. Here it has actually been 
said, with much reiteration, that without regard to other things, 
it is sufficient for us only to know, or it is necessary for us to 
know, on which side the cause of the change lay^ and then we 
should know which of the two has changed — in " value," still 
appearing to mean exchange- value, now of some absolute sort 
(except that we know that these writers associate even the 
term " value in exchange " with value of other kinds). ^ The 
cause in this matter has generally been sought in the labor of 
producing the article (though it might equally well be sought 
in the quantity of the article forthcoming, relatively to the 
number of people desiring it) ; and so it is said that if we knew 
that this labor (or this quantity) has changed in the case of one 
of the articles only, Ave should know which of them alone has 
changed in " value " — supposedly meaning excliange-value. 
But if so, the statement is wrong. For, under the supposition, 
we know nothing about other things ; and so if we knew only 
of such a change in the labor-cost (or in the quantity) of one of 
the articles, we should only know that its cost-value (or esteem- 
value) has changed, not that its exchange-value has changed — 
by " exchange-value " properly meaning only exchange- value 
simply so called, that is, exchange- value in all other things ; 
for the rest are unknown, or left out of account. But as re- 
gards its exchange-value in the one other thing, we know that 
this has changed exactly as we know that the other's exchange- 
value in it has changed, learning both from actual exchanges ; 
and we know these changes just as well whether Ave know 
the changes in their costs of production (or quantities), or not. 
In considering the mutual exchange-values of tAvo things alone, 
Ave are abstracting all other things ; and uoav the only change 
Ave haA^e a right to consider is the change betAveen them, Avhich 
involves both a rise and a fall. 

§ 2. In thus putting aside all other things and abandoning 
the use of them as a common measure, Ave can measure the ex- 
change-value in other things of each of the tAvo objects only by 

1 E. g., J. E. Cairnes, Some leading principles of political economy, 1874, pp. 
13-14. So even AValras, B. 71, pp. 5-6. — Shadwell disposes of this question by 
pronouncing it " puerile," op. cit., p. 103. 



28 THE CORRELATIOX OF EXCHAXGE- VALUES 

the other. As we employ a different measure for each, the rise 
and the fall are not fomid to be in the same proportion. Evi- 
dently if A, after being equivalent to B, rises by one-half, or 
by 50 per cent., in exchange-value in B, so that A exchanges 
for 1|B, then, IJB exchanging for lA, IB exchanges for |A, 
wherefore B has fallen by one-third, or by 33^ per cent., in 
exchange-value in A. 

The relationship just exemplified may be formulated and 
generalized as follows. Here and throughout these pages 
whenever percentage is represented in algebraic formulae by the 
sign p or the like, this refers to the percentage on 1, or to 
percentage expressed in hundredths.^ Now, then, if A, after 
being equivalent to B, rises p per cent, in exchange- value in B, 
so that it commands (1 +j3)B, and has risen to (1 -f p) times 
its former exchange-value in B, then B, now exchanging for 

1 1 « 1 
A, has fallen bv 1 — :^— ; — or —-- — per cent, to 3— -— 

of its former exchange-value in A. Reversely if A, after being 
equivalent to B, falls p' per cent, to (1 — jy)B, B rises by 

1 v' 1 

— 1 or zr^ — 7 iDer cent, to -^ -. times its former ex- 

1 —p' 1 —p' ^ 1 —p' 

change-value in A. Thus the exchange-value of A in B and 
of B in A being in each instance representable at first by 1, we 
find that unity is the geometric mean between the later exchange- 
value of A in B and the later exchange-value of B in A ; for 



1+p' 
1 



l-fj): 1 :: 
and 

^ 1 —p' 

This relationship may be stated thus : mien of two equivalents the 
one rises or falls in exchange-value in the other, the other falls or rises 

2 If the popular form be desired, p referring to integral figures, all the formulae 
which follo^y may be maintained by substituting 100 wherever 1 occurs or is 
understood. £. g., here 100 A, after being equivalent to 100 B, rising p per cent., 
command (100 +_p) B, etc. And always the expression "(l+p) times" must 

be changed to " -.r^o times." It will be seen that the method adopted is the 
simpler. 



BETWEEN TWO THINGS 29 

in exc/iange-value in it so that their subsequent mutual e.vcham/e- 
values are recijirocals of each other, the quantities in which they 
first exchange being taken as units. Or, again, more briefly, the 
rise of the one measured in the other is the inverse of the fall of the 
latter measured in the former ; and reversely (Proposition IX.). 

We may, of course, in our supposition replace B by M, 
representing a sum of money, say one unit. The same rule 
holds, but may be differently worded. Price is the expression 
of the exchange-value in money of the article priced.^ If A 
rises in price from 1.00 to 1.50, its exchange- value in M rises 
in direct proportion, and the exchange-value of M in it falls in 
inverse proportion, M now purchasing only so much of A as 
.66|M previously purchased. The rise of A in price by 50 
per cent., is a depreciation of money in A by 33J per cent. 
Reversely if the price of A falls from 1.00 to .50, or by 50 per 
cent., the exchange-value of M in A has risen by 100 per 
cent., M now purchasing 2A. The jvioe being originally one 
unit, the subsequent price and the subsequent exchange-value of 
money in the article jiriced (^expressed in its original exchange- 
value in ii) are reciprocals of each other (Proposition X.). 

§ 3. In these reciprocal changes it is possible to affirm that, in 
spite of the difference in the proportions, which arises from the 
differences in the measures used, the rise of the one thing in the 
other and the corresponding fall of the latter i)i the former are 
equal (Proposition XI.). A rise of A to double its former ex- 
change-value in B, for instance, and the fall of B to half its 
former exchange-value in A, are equal changes. For the ex- 
change-value of A in B rises from the power possessed by 1 A of 
purchasing IB to that of purchasing 2B ; and if it should fall 
again from this power to that of purchasing IB, this fall would 

^ Economists have not been careful in their use of the term " price," defining 
it indifferently in two distinct ways : (1) as the sum of money given (or asked) 
in exchange for the thing, (2) as the value of the thing in money — and sometimes 
the same writer has given both, e. g., J. S. Mill, op. cit., Vol. I., p. 538 ; .T. Garnier, 
Traite d'economie politique, 1848, 9th ed., 1880, p. 293 ; Macleod, Elements of 
political economy, 1858, p. 39 and Theory of credit, Vol. I., p. 176. The first is 
more conformable to popular usage, and is the best. Price is not the exchange- 
value of the thing in money, but, being the sum of money itself, it expresses or 
measures the exchange-value of the thing in money. 



30 THE COREELATIOJSr OF EXCHANGE- VALUES 

evidently be equal to that rise. In the original variation the 
exchange-value of B in A falls from the power possessed by 
2B of purchasing 2 A to that of purchasing lA. This fall 
is evidently equal to the last mentioned fall of A, and conse- 
quently to that rise of A. In effect, a rise from 1 to 2 and a 
fall from 2 to 1 are equal to each other. 

§ 4. Whenever such mutual changes take place between two ar- 
ticles relatively to each other, their exchange-values in all things 
beside themselves vary in the same proportions relatively to each 
other (Proposition XII.). When A and B are equivalent to 
each other, we have seen that they possess the same exchange- 
value in all other things, whatever this exchange-value may be- 
Then if A rises 50 per cent, in its particular exchange-value 
in B, it is evident that, in relation to all the other things Avhich 
are outside them both, A's exchange-value in all those other 
things has risen 50 per cent, above B's exchange-value in all 
those other things, whatever this exchange-value of B may be ; 
and inversely B's exchange-value in all those other things has 
fallen 33 J per cent, below A's exchange-value in all those other 
things, whatever this exchange-value of A may be. 

What has just been said is not to be said of the exchange- 
value of A in all other things, which would include B, because 
B has changed relatively to A, and so may be a disturbing 
factor ; nor of the exchange- value of B in all other things, 
which would include A, because A has changed relatively to 
B, and so may be a disturbing factor ; wherefore the relation- 
ship between these exchange-values is slightly different. But 
the disturbance caused by B's presence in A's general exchange- 
value is counteracted if A also is included ; similarly in the 
case of B's general exchange-value, if B's presence is also in- 
cluded, wherefore the above relationship is restored. That is, 
when mutual changes take place between two things, relatively to each 
other, their exchange-values in all things vary in the same propor- 
tions relatively to each other (Proposition XIII. ).^ Another 

* In practice, of course, the variation in their exchange-values in all other 
things will be almost the same as their variations in all things, — as will be shown 
later. The Proposition, then, is approximately true of the "general purchasing 
power " of two things. 



BETWEEN TWO THINGS 31 

demonstration of this proposition will be forthcoming when 
we have discovered how to measure exchange-value in all things. 

It follows, of course, that i/ two things (classes) remain un- 
changed relatively to each other, their exchange-values, both in all 
the common other things and in all things, vary, or remain con- 
stant, alike (Proposition XIY.) ; but not necessarily do their 
exchange-values in all other things vary alike. 

Naturally when mutual changes take place between kvo things, 
some change must take place in one or in both relatively to the 
common other things (or relatively to all things, and also to all 
other things) ^ (Proposition XV.). The exchange- values of the 
two things in all the common things (or in all things) are 
yoked together so that, the variation of the one in the other 
being given, and also the variation (or constancy) of either in 
all the others (or in all things), the variation (or constancy) 
of the other in all the others (or in all things) may be easily 
calculated. In general the possible changes may be classified 
into jive principal types: relatively to the other things (1) A 
may have risen, while B has remained unchanged, (2) A may 
have remained unchanged, while B has fallen, (3) A may have 
risen and B fallen, (4) both A and B may have risen, but A 
more, (5) both A and B may have fallen, but B more. Mere 
knowledge that A has risen in B, with or without the percentage 
being known, conveys no knowledge as to which of these pos- 
sible combinations of changes relatively to the other things has 
been effected ; but it does convey knowledge that one or another 
of them must have been. Which of them it is, or exactly what 
the changes are relatively to all the common other things (or to 

^This Proposition, therefore, is true of the general purchasing power of two 
things. Yet the following statement has been made: — "Whilst gold has not 
risen in purchasing power, and silver has not declined in purchasing power, the 
relative value of silver to gold has declined to less than half," A. EUissen, The 
errors and fallacies of bimetallisin, London 1895, p. 12. The author, however, is 
conscious of inconsistency, for he adds : " In mathematics such a problem would 
amount to a preposterous absurdity ; but in the case before us it is only too true." 
It is tenable only by an equivocation, the author having treated "purchasing 
power " like " value " in general, and then not having distinguished between its 
different species. He means that silver has not fallen in purchasing power proper 
and gold has not risen in cost-value (or esteem-value) — perfectly simple and un- 
connected statements, not deserving to be so expressed as to involve contradiction. 



32 THE CORRELATION OF EXCHANGE- VALUES 

all things) must be sought — not by trying to find the causes, or 
changes in the costs of production (or in the quantities) of either 
or both, or of all the other things,'' but — by direct measure- 
ment either of A's or of B's exchange-value in all the other 
things (or in all things), or of both, which measurement must 
take account of all the particular exchange-values of either or 
each in all the other things singly, and thereby also of the par- 
ticular exchange- values of all the other things in either or each, 
of these two. 

II. 

§ 1. Let us now pay attention to the first and the second,, 
as the simplest, of the above five possibilities when the other 
things are also taken into consideration. Let us suppose that, 
A rising or falling in exchange-value in B, B retains its ex- 
change-value unchanged relatively to all the other things, and 
that these other things all remain without change relatively to 
one another and to B. Now, then, if A rises in exchange- 
value in B, it rises also in exchange-value in every one of the 
other things, that is, in every other thing beside itself, in the 
same proportion ; and, B ijjso facto falling in exchange-value 
in A, so does every one of the other things fall in exchange- 
value in A, and in the same proportion as B, that is, in inverse 
proportion to A's rise in B. Thus if A rises 50 per cent, in 
B, it rises 50 per cent, in C, in D, and so on throughout all the 
other things ; and as B falls 33J per cent, in A, so do C and 
D and the others, each singly, fall 33|^ per cent, in A. And 
the reverse is the case if A falls in B. Then, like B, all the 
others rise singly in A in inverse proportion to A's fall in 
B. Thus we have the general law : A variation of one thing in 
exchange-value in the same proportion relatively to every other thing 
involves an opposite variation in inverse proportion of every one 
of the others relatively to it (Proposition XVL). 

^ These are subjects that must be investigated in order to measure variations of 
cost-value, or of esteem-value. Naturally, different kinds of value must be 
measured in different ways, and the ways required for measuring variations of 
other kinds of value are not the ways required for measuring variations of ex- 
change-value. 



Bf^TWEKN ONE THIXa AND ALL OTHERS 33 

When A thus varies, say rising or falling 50 per cent., in re- 
lation to every one of the other things, all these remaining un- 
changed amongst themselves, as they must do under the first 
supposition, it is evident that A has varied in the same propor- 
tion, rising or falling 50 per cent., in exchange-value in all 
these other things together. In other words, a variation in the 
particular exchange-values of any one thing in the same pro- 
portion in all other things singly is a variation in its general 
exchange-value in the same proportion in all other things col- 
lectively. Or more briefly, lohen all the particular exchanr/e- 
values of a thing in other things vary alike, their common vari- 
ation is the variation of its general exchange-value in all other 
things (Proposition XVII.). 

It may be noticed that absolutely all the particular exchange- 
values of a thing cannot vary ; for its particular exchange- 
value in itself (or in its mates within its class) never varies (ac- 
cording to Proposition II.). Hence the need of confining the 
statement to the particular exchange-values in other {i. e., in 
other kinds of) things. 

§ 2. But when B and C and D and every other thing have, 
say, fallen by 33^ per cent, in exchange- value in A, it is equally 
plain that B and C and D and every other thing have not fallen 
so much in exchange-value in all other things ; for they have 
not fallen relatively to one another. All A's particular ex- 
change-values in other things have risen in the same propor- 
tion, namely by 50 per cent., and therefore A's general ex- 
change-value in all other things has risen in that proportion. 
But only one particular exchange-value of B, of C, of D, etc., 
has fallen, while all the other particular exchange- values of B, 
of C, of D, etc., have remained unchanged; therefore their 
totals of particular exchange-values, that is, their general ex- 
change-values in all other things, have been less altered. Of 
course the reverse takes place in the case of a fall of A. If A 
falls in exchange-value in every other thing in the same pro- 
portion, it falls in exchange-value in all other things in that 
proportion ; and every one of the other things rises in exchange- 
value in it alone in inverse proportion without rising in any- 
3 



34 THE CORRELATION OF EXCHANGE- VALUES 

thing else^ and therefore every one of them rises in exchange- 
value in all other things in a proportion smaller than the 
inverse. Thus we have the general law : A variation of any 
one thing in exchange-value in the same jproportion relatively to 
all other things involves an opposite variation of every one of the 
others in exchange-value in all other things to a smialler extent than 
the inverse proportion (Proposition XVIII.). 

And evidently, in such cases, the extent of the opposite varia- 
tions will he smaller and smaller the more numerous are the other 
things (Proposition XIX.). We perceive this by supposing at 
first the existence of only three articles and then extending the 
number. Let A, as before, be the article which rises in ex- 
change-value in every other single thing, say by 50 per cent. 
Then, as all the other things are in the same box, having the 
same exchange-values, we need to follow the fortunes of only 
one of them, say B. Now with only three articles,^ B was at 
first equivalent to A and to C, and after the rise of A it is 
equivalent to f A and to C ; with four articles, it Avas equiva- 
lent to A, to C and to D, and later is equivalent to |A, to C 
and to D ; with five, it was equivalent to A, to C, to D and to 
E, and later is equivalent to f A, to C, to D and to E ; and so 
on. It is plain that the larger the number of other things, the 
larger is the number of its particular exchange-values which B 
retains unchanged, and the comparatively smaller and less ap- 
preciable becomes the loss of its exchange-value in A — or 
smaller and less appreciable the loss of its purchasing power 
over A compared with its purchasing power over the other 
things which remain unchanged ; — and consequently, the whole 
being enlarged, the smaller is the fixed loss in one member 
relatively to the remainder. With very many articles, a small 
change in A relatively to the rest, may be a practically inap- 
preciable change in B and the others relatively to all other 
things as a whole.^ But theoretically, in any such case, how- 

1 We must begin with tlaree in order to get two others, so as to make the 
plural. If, however, we started with only two things, having only one other for 
"all the others," the opposite variation would be exactly in the inverse propor- 
tion, this condition reducing to the condition in Propositions IX. and X. This is, 
for really other things, the unattainable limit of greatness in the opposite variation. 

2 Of course in all this passage the sizes of the other articles are supposed to be 



BETWEEN OXE THIXG AND ALL OTHERS 35 

ever small the change of A, there is some change in B and the 
others relatively to all other things as a whole.^ The reverse 
is equally plain if we suppose A to fall. 

§ 3. What is said of the correlation of exchange-values in 
the case of things in general, is of course true when one of 
them is money. To reinterpret these relations in terms of prices, 
is a simple matter. (1) If money rises in exchange-value in 
every other article by p per cent., it rises in exchange-value in 
all other things by |3 per cent. But its rise in exchange-value 
in every other article by j) per cent, is the same as a fall of 

P 
every one of these in price by z per cent. Therefore a 

7) 

uniform fall of price by ~ per cent, is a rise of money in 

exchange-value in all other things by p per cent., that is, in in- 

p 

verse proportion. And while every article has fallen by z 

per cent, in price, that is also, by r— - — per. cent, in exchange- 
value in money, every article has fallen to a much smaller ex- 
tent in exchange-value in all other things.* And similarly in 
the case of a fall of money or a rise of prices. Yet we some- 
given. If we increase their number simply by breaking them up into parts, the 
extent of tlie opposite change in their exchange-values is not thereby aifected. 
The sizes of articles, or classes, will be treated of later. 

* If it be theoretically permissible to suppose an infinite number of equally 
larger other things, then the opposite change, no matter how much A or any finite 
number of things finitely rise in every one of the common other things, the op- 
posite variation of every one of these in exchange- value in all other things is, 
even theoretically, infinitely small, or nil, that is, there is no variation in their 
exchange-values in all other things. This is the unattainable limit of smallness 
in the opposite variation. 

* It is a mistake to say that the "general value" of these articles has not 
changed, as said by J. Prince-Smith, Valeur et monnaie, Journal des Economistes 
Dec. 1853, p. 373. But if we purposely ignore the exchange-value of money on 
the ground that money is only an intermediary in exchanges of goods, it is true 
to say that the general exchange-value of these articles in all other, or in all, 
articles except money, has not changed. We should remember, however, that 
money is not an intermediary merely in contemporaneous exchanges, but that 
frequently a long interval passes between the time when we get money for our 
property and the time when we get other property for our money, so that the in- 
tervening state of the exchange-value of money is very important. Still the im- 
portance seems to be sui generis. 



36 THE CORRELATION OF EXCHANGE- VALUES 

times hear such faulty expressions as this : If, prices rising by 
50 per cent., money depreciates by 33^ per cent., commodities 
have appreciated by 50 per cent.; or conversely. It is true 
that commodities, each singly, have appreciated by 50 per cent, 
in money ; but to say simply that commodities have appreciated 
50 per cent, means that they have risen 50 per cent, in ex- 
change-value simply, which is not true.^ Again (2) if money 
rises by p per cent, in exchange-value in one article alone 
while retaining unaltered its exchange-value in every other 
article, this is only another way of saying that the price of 

%> 
that one article has fallen by :j per cent., while the prices 

of all other articles has remained unchanged. Then the ex- 
change-value of that one article in every other thing, and in 

V 
all other things has fallen by z per cent.; and the exchange- 
value of every other article, as of money, in that one thing, 
has risen by p per cent., but the exchange-value of money, and 
of every other article, in all other things, though it has risen 
somewhat,^ has by no means risen to that extent. 

III. 

§ 1. A corollary from the preceding laws is this : There 
cannot be a variation in the exchange-value of one thing alone 
(Proposition XX. ).^ There can, of course, be a rise or a fall 
in exchange-value of one thing alone, since one thing may rise 
without anything else rising, and one thing may fall without 
anything else falling. But the moment one thing alone so rises 
or falls in exchange-value, all the others inversely fall or rise. 

^ Another similarly faulty expression is that in the case of a rise of prices 
" values " have risen ; and conversely. It is only values in money, or values 
expressed in money, that have changed exactly as the prices. Of course on the 
stock-market, where the only values thought of are money -values, this expression 
is natural. 

° As correctly said by Prince-Smith, op. cit., p. 373, and by Jevons, B. 15, p. 
20. 

^ So Walras, Elements, 1st ed., pp. 167-168, 2d ed., p. 456. This is very differ- 
ent from what is referred to in Note 1 in Sect. I. of this chapter. Here he is 
speaking of exchange-value ; there, what he had in mind was esteem-value. 



INVARIABILITY OF ONE THING ALONE 37 

Hence it is a solecism to speak simply of a change of one thing 
alone in exchange-value. There may be a change of one price 
alone ; but as we have just seen, this is not a change in the ex- 
change-value of the one article alone. Furthermore there may 
be a change in the use-value, in the cost-value, and perhaps even 
in the esteem-value, of one thing alone. But here is only one 
more reason for distinguishing use-value and cost- value and 
esteem-value from exchange-value. Likewise it is conceivable 
that a cause may act upon one thing alone to make its price 
change, or its use-value change, or its cost-value or esteem-value 
change, without affecting the price, use-value, cost-value or per- 
haps (although this is questionable) even the esteem-value, of 
anything else. It is not possible, however, for a cause to act 
upon one thing alone so as to render it of greater or smaller ex- 
change-value, since this cause will also act upon all other things 
to render them of smaller or greater exchange-value in this thing 
and thereby, to some extent, in all things.^ 

§ 2. What is here said shows the wrongness of a doctrine 
sometimes taught concerning "value," supposedly in the sense 
of exchano;e-value. This is that when a sensible rise or fall 
takes place in the value or price of one thing alone (or in the 
value of money, and therefore in all prices in the opposite di- 
rection), the rest remaining tolerably steady relatively to one 
another, there is reason for holding that the change is altogether 
in the one thing and not at all in the others — that is, that the 
one thing alone has changed in " value," and the others have 
remained unchanged in " value." The reason assigned is that 
it is easier to explain these phenomena by supposing one cause 
affecting this one thing than by supposing causes affecting the 
other things, which causes would have to be as numerous as the 
other things ; or, in other words, that the hypothesis invoking 

2 Hence there is impossibility iu tlie assumption made by H. Fawcett (follow- 
ing J. S. Mill, op. cit., Vol. I., pp. 539-540) in his Manual of political economy, 
18(33, 6th ed., 1883, p. 314, at the commencement of his inquiry into " the causes 
which regulate the price of commodities," namely that a variation of price " is 
always supposed to be produced by something which affects the value of the com- 
modity and not the value of precious metals"; for by "value" he meant ex- 
change-value. Fawcett has been followed by A. S. Bolles, Chapters in political 
economy, 1874, p. 56. 



38 THE COEEELATIOX OF EXCHANGE- VALUES 

one cause is far the simpler and more probable — -just as the 
Copernican theory is simpler and more probable than the Ptol- 
emaic.^ 

The statement made in this reason assigned for the doctrine 
in question is in itself not incorrect. In default of data for 
finding the cause or causes directly, or in our impatience at the 
delay required for such a direct investigation, it is easier for 
us to suppose, and we are inclined to believe, that there is only 
one cause affecting, or, as is sometimes carelessly said, residing 
in, the one thing whose exchange-value in all others is most 
prominently changing.* But to take this view of the matter as 
a reason for supposing a change in exchange-value only in the 
one thing, is absolutely incorrect. We can — it being supposed 
all along that we know the particular variations — calculate with 
mathematical precision, wdien we have fully developed the 
theory of exchange-value, almost the exact variation in the ex- 
change-value (in all other things, and also in all things) of the 
one thing and almost the exact variations in the (similar) ex- 
change-values of all the other things. There is no occasion for 
employing the law of probabilities at all. 

The explanation of this doctrine itself — and of the use of the 
law of probabilities with regard to the causes as a reason for 
applying the same to the changes of " value " — is the retention 
in the idea of '" value " of the ideas of cost- value and of 
esteem-value. In the case of cost-value, and perhaps also of 
esteem-value, it would be possible for the change to reside only 
in the cost-value, or in the esteem-value, of the one thing ; 
and accordingly here, in default of knowledge to the contrary, 
it is more probable, it is a more likely hypothesis, that the 
change has taken place in the cost-value, or in the esteem-value, 
of the one thing alone. This is because, at least in the case 
of cost-value, this value is wholly dependent upon tlie labor 
required for producing the thing, and so is measured by that 

' Eicardo, Works, p. 13; A. Cournot, Becherches sur les principes mathe- 
matiques de la theorie de la richesse, 1838, pp. 18-19 ; the latter followed ("with 
addition of the simile) by C. Gide, Principes d'economie politique, 1884, pp. 
81-83. 

* Thus this hypotliesis ia regard to the cause, but not in regard to the change 
in value, was used by Jevons, B. 22, pp. 19-20, B. 24, p. 156. 



INVARIABILITY OF ONE THING ALONE 39 

labor, that is, is measured by a measurement of its cause. Now 
if it should really happen that one thing alone rose or fell in 
cost-value, the relation of its cost-value to the cost-value of all 
the other things would have varied, and, if it rose compared 
with them, they would have fallen compared with it ; and if, 
as is sometimes the case, the exchange-value of the thing varies 
with its cost- value, its exchange- value would rise in other 
things and theirs would fall in it (and consequently in exchange- 
value simply). Here then we should be reaching a right con- 
clusion by means of the law of probabilities if we supposed the 
change of cost-value to be solely in the one thing ; but we 
should be violating the assumed conditions if we concluded 
either that the relative cost-value of the thing compared with 
the others has alone changed and not also their relative cost- 
values compared with its, or that the exchange-value of the one 
thing (meaning its exchange-value simply, /. e., in all other 
things, or in all things, or compared with the exchange-values 
of the other things) has alone changed and not also the ex- 
change-values of the other things (meaning their exchange- 
values simply, /. ('., in this thing and in all other or in aU 
things, or compared with this thing's exchange-value). The 
very data assumed require these correlative changes both in re- 
lative cost-values and in exchange-values — for all exchange- 
value is relative.'^ 

IV. 

§ 1. Another corollary from the above-given laws is the 

already mentioned distinction between exchange-value in all 

* There is, therefore, no need of speaking of "relative exchange-value." In 
fact, the very term "exchange" in this compound term has the meaning of 
" relative," so that " exchange-value " is similar to " relative value." Sometimes 
exchange-value is relative cost-value ; always it is relative esteem-value. In the 
course of time a thing retains the same exchange-value if it retains the same rela- 
tive esteem-value, that is, whether its esteem-value vary or not, if it varies or not 
just as the esteem-values of all other things on the average vary or not. Or in 
the course of time its exchange-value varies if its relative esteem-value varies, 
that is, if its esteem-value varies compared with the average of the esteem-values 
of all other things, for instance rising (1) if it rises more than they do, or (2) if 
it rises while they are stationary, or (.3) if it rises while they fall, or (4) if it is 
stationary while they fall, or (5) if it falls less then they do — in the same five 
typical relations we have already noticed in another connection. 



40 THE COERELATION OF EXCHAXGE-VALUES 

other things and exchange-value in all things. For the law 
that if one thing alone rises in exchange- value in every other 
thing to the same extent, it rises exactly to that extent in 
exchange-value in all other things, is perfectly evident. And 
also the law is equally plain that every one of those other 
things has sunk somewhat in exchange-value in all other 
things. Putting these two laws together, we see that when a 
thing rises a certain percentage in exchange-value in all other 
things, as it rises to that extent in things that have themselves 
sunk somewhat in exchange-value in all other things, it has 
not risen quite that percentage in exchange-value after all, but 
to a somewhat lesser extent. The performance is comparable 
with the rise of a body above a plane which is depressed by 
the force raising the body ; in which case the measurement of 
the rise of the body in relation to the plane gives a greater re- 
sult than the true rise. Similarly in the reverse case of the fall 
of a body by which the plane from which it falls is repelled 
upward, the measurement of the fall of the body from the plane 
also gives too great a fall. So a fall of one thing in exchange- 
value in all other things, being a fall which raises their ex- 
change-values somewhat, is not so great a fall really as it ap- 
pears to be, judged by comparison with them only. 

The explanation of the difference is obvious. In treating 
of the exchange-value of one thing in all other things we are 
using all other things as the standard ; but then in treating of 
the exchange-value of any one of the other things also in all 
other things we are using a different standard, for this standard 
excludes the latter thing which was included in the former 
standard, and includes the former thing, which was excluded 
from that standard. These two standards, then, are shifting 
standards ; but together they include all things. Thus when 
we perceive that a thing which has risen a certain extent in 
exchange-value in all other things has not risen quite so high 
in exchange-value, by this exchange-value we mean an ex- 
change-value measured in a standard in which not only all the 
other things are included, but also the thing itself of which the 
altered exchange- value is being measured. Thus this exchange- 



TWO KINDS OF GENERAL EXCHANGE-VALUE 41 

value is exchange-value, not in all other things, but in all 
things. And it is the superior and final exchange-value, the 
one more than any other appearing to deserve the single term 
exchange-value. We might even be tempted to call it " abso- 
lute exchange-value," except that there are metaphysical and 
logical objections to the use of this term. These objections 
could not hold against calling it " universal exchange-value." 
But perhaps it is as well not to give it any name. 

The simple term, then, exchange-value, or general exchange- 
value, is slightly ambiguous. The same ambiguity extends to 
the related terms, appreciation and depreciation, and the like. 
Thus a thing which appreciates a certain percentage in ex- 
change-value in all other things, does not appreciate quite so 
much in exchange-value in all things. Yet almost all writers 
on the subject have measured appreciation and depreciation, or 
in general any variation in exchange-value, in the former way 
only. Thus it is commonly asserted that if all other things 
remain unchanged amongst themselves, including money, the 
change of one thing relatively to any one of them, and so its 
change in price, exactly measures its change in general ex- 
change-value ; ^ or again that if all prices change alike (or on 
an average) to a certain extent, money has changed to the in- 
verse extent in exchange-value simply.^ Such statements are 
only half true. They are true of the variations in exchange- 
value in all other things ; but they are not true of the varia- 
tions in exchange-value in all things. 

§ 2. The differences between these two kinds of general ex- 
change-value and the laws common to them both, may be briefly 
surveyed. It is evident that a thincf s exchange-value in all 
things always varies less than its exchange-value in all other things 
(Proposition XXL). For, given the deviation of a thing's ex- 

1 JS. g., J. S. Mill, op. cit., Vol. I., pp. 539-540. 

- So Priuce-Stnith, op. cit., p. 373. J. S. Mill says that, caeteribus paribus, 
prices vary directly, the exchange-value of money inversely, with the quantity of 
money, op. cit.., Vol. II., pp. 16-17. It is usual, after getting a method or formula 
for measuring the variation of the average of prices, simply to invert this as the 
measure of appreciation or depreciation of money ; so Levasseur, B. 18, p. 195; 
Jevons, B. 22, pp. 5.3-54, 58 ; Drol>isch, B. 29, p. 39, B. .'^O, p. 149 ; Lelir, B. 68, p. 
40; NichoLson, B. 94, pp. 317-;U8; Lindsay, B. 114, p. 26. 



42 THE CORRELATIOX OF EXCHANGE- VALUES 

change-value in all other things, as its exchange-value in itself 
never varies, when this unchanged particular exchange-value of 
itself in itself is added to the standard by which its deviation is 
measured, it must have the effect of lessening the deviation 
somewhat. But it cannot annul the deviation, since there is 
some particular variation as before, which must still be counted. 
Therefore the one general exchange-value of a thing cannot vary 
loithout the other varying also (Proposition XXII.). And, of 
course, the lessening just shown cannot deflect the deviation in 
the opposite direction ; wherefore the two kinds of general ex- 
change-value always deviate from constancy in the same direction 
(Proposition XXIII.). Further, as we have seen that when 
one thing alone rises in exchange-value in all other things, the 
others fall somewhat, and less the more numerous they are, and 
conversely, so we now see that there are two rises of the one 
thing and consequently also two falls of the other things, and 
conversely. It is now plain that in both cases, the divergence 
between the two deviations of the two kinds of general exchange- 
value is smaller the more numerous are the articles (Proposition 
XXIV.). For the divergence is determined by the difference 
between the whole number of things minus one thing and the 
whole number of things compared Avith the whole number of 
things, and this comparative difi'erence is smaller the larger the 
whole number of things. Thus, for example, if we know the 
variation of a thing in exchange-value in all other things (as 
when conditions satisfy Proposition XVII.), then we know 
that its variation in exchange-value in all things is almost as 
great, if the other things are very numerous (or very much 
larger than the one thing, or than its class). Similarly, given 
the number of (equal) articles, the divergence between the two devi- 
ations is always in the same proportion (Proposition XXV.). 
For here the comparative difference is the same in all the 
measurements. We shall later find formulse from which, in 
some cases, the exact proportions can be derived, if desired^ 
and a very simple proposition expressing them. 

§ 3. Upon these laws of variation and divergence follows a 
law of constancy and identity. It is plain that if there is no 



TWO KINDS OF GENERAL EXCHANGE- VALUE 43 

variation of a thing's exchange-value in all other things, there 
can be no variation of the thing's exchange-value in all things, 
since nothing is added in the latter case to make a variation not 
existing in the former. Nor, reversely, if there is no variation 
of a thing's exchange-value in all things, can there be a vari- 
ation of the tiling's exchange- value in all other things, because, 
if an element of variation existed in the latter case, it would 
have existed in the former also. In other words, constanci/ of 
the one kind of general exeluinr/e-cahie j)0ssessed by anything is 
constancy of the other (Proposition XXVI.). That is, again, 
the two kinds of general exchange-value coincide and are identical 
when either of them is constant. Another demonstration of this 
theorem may be offered as follows : If changes in a thing's 
particular exchange-values occur so that its exchange-value in 
all other things rises, its exchange-value in all things rises, but 
less ; and if such changes occur so that its exchange-value in 
all other things falls, its exchange-value in all things falls, but 
less ; therefore if we suppose these changes on each side to be 
smaller and smaller until the change in the thing's exchange- 
value in all other things is so small as to be inappreciable, its 
exchange- value in all things, always being even smaller, must 
be still less appreciable, and when the former becomes zero, the 
latter must also be zero. 

On account of this coincidence of the two kinds of general 
exchange-value at rest, we are justified in subsuming them under 
the same term ; and whenever we have occasion to speak of 
constant general exchange-value, or simply of constant exchange- 
value, there is no need of distinguishing between the two kinds, 
which separately exist only when there is variation.^ 

' Beginning at a given period, we see that if there is a variation of a thing's 
general exchange-value, this splits up into two variations from the original co- 
incident positions of the two kinds of general exchange-value ; but if there had 
been constancy of the thing's general exchange-value, there would have been no 
such splitting, and the two kinds would not have manifested their distinctness. 
Now when such a variation and such a splitting have occurred, if we start over 
again from the second period as a new starting point, the same phenomena may 
repeat themselves ; that is, if from this starting point thei-e is constancy, the 
general exchange-value remains undivided, but if there is variation, it divides 
again into two. And so on from any later period as a new starting point. In 
every case the two kinds of general exchange-value at the second period, com- 



44 THE CORRELATION OF EXCHAXGE-VALUES 

V. 

§1. Constancy of general exchange- value may happen under 
two different conditions of things. The first is the simplest 
and most obvious. It is evident that if all the particular ex- 
change-values of a thing in all other things remain constant, its 
general exchange-value in all other things and, since its ex- 
change-value in itself must remain constant, its general ex- 
change-value in all things remain constant. This condition 
takes place when there are no changes whatsoever between the 
particular exchange-values of any thing in any other amongst 
all things ; so that the general exchange-value not only of the 
thing in question, but of everything else, is without change. 
Thus, if all things retain the same exchange-values relatively to 
one another, the general exchange-value of any one thing is con- 
stant (Proposition XXVIL). 

The second condition is not quite so obvious, but is equally 
demonstrable. For evidently it is possible, when one thing 
changes alike in exchange-value in all others except one, for all 
the rest beside these two to remain unchanged in exchange- 
value in all other things, and consequently in all things, pro- 
vided that the other one thing change in respect to them in a 
manner to compensate for the change of the first one. And 
consequently, any two compensating for each other, or several 
compensating for one or for several, or many for many, it is 
possible for one thing to retain the same general exchange- value, 
in spite of changes in any two or in any higher number of others 
up to all others. 

§ 2. Here we meet with an error which would be fatal in 
our subject. A few economists have asserted that the general 
exchange-value of anything can remain unchanged only under 
the first condition, and as that condition is never fulfilled in 
the actual world except over the briefest of intervals, they 
have denied even the possibility of any one thing, or in especial, 
money, retaining the same general exchange-value through the 

pared with the first, may have remained the same or have divided ; but in every 
case the two kinds at the first period, to be compai'ed with the second, are taken 
as identical. 



THE POSSIBILITY OP CONSTANCY 45 

course of time^ — and one writer has even rejected the expression 
" stable value " as a collocation of words without meaning, like 
" triangular square." ^ The same position has been taken by 
Roscher concerning general purchasing power ^ — with better 
reason, as we shall see, for one sense in which this term is 
used. Now if this view were correct in regard to exchange- 
value, that is, if the constancy of general exchange-value under 
any changes of particular exchange-values is even theoretically 
impossible, or inconceivable, it would be absurd to speak of 
a greater or less variation of a thing's general exchange- 
value through the course of time, since this expression always 
has reference to its variation from a supposed constant position, 
and if this has no possible existence, the magnitude of the vari- 
ation has no possible existence — nay, the variation itself could 
not possibly exist, and the general exchange-value of a thing 
at one time would be wholly incommensurable with its general 
exchange- value at another time, as indeed has been maintained 
by some extremists.^ Then not only to measure a thing's con- 
stancy in general exchange- value would be an idle endeavor, 
but also to measure its variation in general exchange- value 
would be meaningless. And the statement frequently made 
by the very persons who deny the possibility of constancy of 
general exchange- value, or the possibility of measuring it, that 
the desirable feature in money is that it should vary as little as 
possible in exchange-value, would be an absurdity ."^ 

1 Galiani, Delia vioneta, 1750, ed., Custodi, Vol. II., p. 11 ; ]\IcCull()ch, Polit- 
ical economy, p. 213 (Note to Wealth of nations, p. 439) ; Prince-Smith, op. cit., p. 
373 ; Macleod, Theory of banking, Vol. I., p. 69, Theory of credit. Vol. I., p. 176 ; 
Walras, Elements, 1-st ed., pp. 167-16S, 185, 2d ed., p. 456; Martello, op. cit., pp. 
402-403. — Rossi seems to have had the same idea when lie said there can be no 
measure of value because as a standard it sliould be invariable and as a value it 
must be variable, Conrs d'economie politique, 1840, Vol. I., p. 1,50. 

2 F. Ferrara, Introduction to Martello's work, p. CXLIX. 

3 B. 32, § 127. 

*£". g., A. Held maintained that we cannot properly say that in a financial 
crisis the exchange-value of money is greater than it was, but only greater than it 
would have been had no crisis arisen, iVbc/i eimmal iiber den Pre/is des Geldes. 
Ein Beitrag znr Mi'inzfrage, Jahrbiicher fiir Nationaloekonoraie und Statis- 
tik, Jena 1871, p. 328. He might as well say that when a yard-stick expands 
under the iniluence of heat, it is not longer than it was, but only longer than it 
would have been had it not been heated. How it is we can compare a value with 
what it would have been, but not with what it was, is not explained. 

^ Among the writers mentioned in Notes 1 and 2 only Ferrara and Martello 



46 THE COEEELATION OF EXCHANGE- VALUES 

§ 3. But the reasoning by which is maintained the denial of 
the possibility of a constant exchange- value through variations 
of particular exchange-values, is the plainest of fallacies. It is 
based on our Proposition XX. The exchange-value of a thing 
is affected and altered if any one other thing varies in exchange- 
value. Therefore, it is concluded, its exchange-value must be 
still more affected and altered, if two other things, if three, if 
many, vary in exchange-value — that is, if any other thing or 
things vary in exchange-value. Now it is plain that the greater 
the number of other things that vary in exchange-value, the 
more the exchange- value of the thing in question is affected; 
but as soon as it is affected by hco or more other influences, the 
possibility arises that it may be affected in opposite directions, 
so that the alteration of its general exchange-value, instead of 
being increased, may be diminished, and furthermore, that it 
may be reduced to zero by the opposite influences exactly 
counterbalancing each other. 

This may be rigorously demonstrated as follows. Suppose 

at first 

M == A =c=B^^ 0=0= ....... 

and later A rises in exchange-value in all the others, so that 
M===-A==B=c=C=o= ; 

X 

then M has fallen somewhat in general exchange-value. Again, 
instead of this rise of A, suppose B falls, so that 

M==A = 2/B===C=c= , 

(in each of these expressions x and y being greater than unity); 
then M has risen somewhat in general exchange-value. Now 
it is evident that if these two changes take place together, so that 

M^iA^2/B==C^ , 

are guiltless of this inconsistency. Walras tries not only to measure general ex- 
change-value, but also to regulate the general exchange-value of money with a 
view to its constancy. Several other economists, who have, for various reasons, 
denied that we can measure exchange- value, or that money is a measure of it, 
have fallen into similar inconsistency. 



THE POSSIBILITY OF CONSTANCY 47 

then the general exchange-value of M is influenced to a fall by 
the one change and by the other to a rise, and it can be that, 
the amount of the first change being fixed, the influence of the 
second change may outweigh its influence, conducing to a fall, 
or the influence of the second change may be insufficient, per- 
mitting a rise. It is also evident, in accordance with the prin- 
ciple of continuity, that between these higher and lower influ- 
ences of the variable second change, compared with the fixed 
influence of the first, there must be a point at which that in- 
fluence exactly counterbalances this influence, so that, the two 
influences neutralizing each other, or having the same effect as 
if neither existed, the general exchange-value of M remains 
unchanged — just as two opposite and equal forces will keep a 
body at rest. 

Now the rise of A in exchange- value in all other things is 
the inverse fall of M's particular exchange-value in A, and the 
fall of B in exchange-value in all other things is the inverse 
rise of M's particular exchange-value in B. Therefore, the 
general exchange-value of anything may remain unchanged in 
spite of a imrlation in one of its particular exchange-values {or an 
inverted variation in the general exchange-value of one other thing), 
if this be compensated by an opposite variation in another of its 
particular exchange-vcdues (or an opposite variation in the general 
exchange-value of another thing) (Proposition XXVIII.). 

Furthermore, it is obviously not necessary for the counter- 
balancing to be done by a variation of one other thing alone. 
The compensation for the variation of one thing may be effected by 
variations, in the opposite direction, of many others, or even of all 
the others (Proposition XXIX.). If done by more than one 
other, it is, however, plain that the variations of the others in- 
dividually must be smaller than the variation of only one other, 
because their combined influence constitutes the compensation. 
If, for instance, the rise of A were already compensated by the 
fall of B, then a fall of C would disturb the balance and itself 
require to be compensated either by the rise of something else or 
by a still higher rise of A. Thus a great rise of A may be 
compensated by two lesser falls, equal or unequal to each other, 



48 THE CORRELATION OF EXCHANGE- VALUES 

of B and C ; consequently, in general, any rise of one thing, by 
any number of falls of other things, and conversely. And, as 
the compensating influence must be distributed over all the 
other changes, it is plain that, the sizes of the things (or classes) 
being given, the greater the number of the compensating variations 
the smaller must they be individually or on an average (Proposi- 
tion XXX.). 

Also it is plain that, a certain variation in one direction being 
compensated by one or many variations in the opposite direction, 
it is possible for still another variation in either direction to be 
compensated by still another or other variations in the direction 
opposite to it, and so on indefinitely, all at the same time, the one 
thing in question remaining unchanged in general exchange- 
value so long as all variations in one direction are compensated, 
in pairs, or in combination, by variations in the opposite direc- 
tion (Proposition XXXI.). 

§ 4. Now when the one thing we have in mind — M in the 
above example — remains constant in general exchange- value 
under changes in two or more other things, it is plain that all 
the other things that have not varied relatively to it — the C, 
etc., in the above example — have also remained constant in 
general exchange-value (in accordance with Proposition XIV.). 
Consequently, the constancy of any one thing in genercd exchange- 
value is uiutfected. by the number of other things likewise constant 
in general exchange-value, and in any calculation it ought to be 
indifferent whether we include them or not (Proposition 
XXXII. ). Evidently it is only things that vary in general 
exchange-value that can exert influence to cause a variation in 
the general exchange-value of anything that would otherwise 
be constant.^ 

But with reo-ard to variations a distinction is to be drawn. 
The things ivhich have varied in certain proportions relatively to 
the thing or things remaining constant have varied in those same 
proportions in their general exchange-value, not in all other 
things, but in all things (or in all but any or all of the constant 

^ Of course if money has varied in general exchange-value, the number of 
things that are constant in price affects the condition of anything otherwise con- 
stant, because these things have varied in general exchange-value with money. 



THE POSSIBILITY OF CONSTANCY 49 

things) (Proposition XXXIII.). Thus, while the variation of 
anything in general exchange-value in all other things is af- 
fected by the number of other things constant in general ex- 
change-value, the variation of anything in general exchange- 
value in all things {or in at least all the variable things, including 
itself) is not affected by the number of other things remaining con- 
stant in general exc/iange-value (Proposition XXXIV.). The 
latter part of this proposition is apparent, since the other things 
that are constant in general exchange-value do not affect the 
general exchange-value of the one thing above used, by com- 
parison Avith which the variation of the one thing may be 
measured, and this variation may be measured as well by com- 
parison with any one of the constant things as by comparison 
with the one first used. The distinction, then, is needed in 
order to carry out Proposition XXIV., which shows that the 
relation between the two variations in general exchange-value 
of anything is aifected by the number of other things ; for then 
the number of other things must affect the two variations dif- 
ferently, and if the number of other things that are constant in 
general exchange-value does not affect the variation of any 
one thing in general exchange- value in all things, it must affect 
the variation in general exchange-value in all other things. 
More particularly, the distinction is evident, because in the 
calcidation by which the constancy of the constant things is 
determined has entered the variation of the exchange-value of 
the thing in question that has varied, but this variation is ex- 
cluded from the calculation by which its variation in exchange- 
value in all other things is determined. It is plain that only 
if we found that money, for instance, remained constant in ex- 
change-value in all other things but one, this one being left out 
of the calculation, would a variation of this one relatively to 
money (indicated by its price-variation) measure the variation 
of its exchange- value in all other things. And what would 
not affect the exchange-value in all other things of a thing that 
varies is, not something remaining constant in exchange-value 
in all tilings, but something remaining constant in exchange- 
value in all but that one thing, that one thing being left out of 
4 



60 THE CORRELATION OF EXCHANGE-VALUES 

the calculation of the constant thing's constancy just as it is 
left out of the calculation of its own variation in exchange- 
value in all other things. In exchange-value in all things 
everything is measured by the same standard, and so has the 
same variation whether it be directly measured in relation to 
all things or whether it be measured in relation to something 
whose position in relation to all things has already been deter- 
mined. But in exchange-value in all other things the standards 
are different in every measurement, and therefore the results 
do not fit and dove-tail into one another as they do in the other 
case. 

The last Proposition may be altered into this : If in measuring 
the exchange-value of money in all things [including itself) we find 
it has varied to a certain extent, then it ought to be indifferent in the 
calculation whether we include or exclude anything lahose price has 
varied inversely (Proposition XXXV.). In other words, for 
instance, if in measuring money we find that the general level 
of prices, including its own, which never varies, has risen so as 
to be r times what it was before, wherefore the exchange-value 

in all things of money has fallen by—, then anything whose 

price has risen so as to be r times what it was before ought to 
be indifferent in the calculation by which the result for money 
was obtained ; for that thing has remained constant in exchange- 
value in all things. Furthermore also, If in measuring the ex- 
change-value of money in all other things (i. e., in all commodities) 
we find it has varied to a certain extent, then it ought to be indif- 
ferent in this calculation whether we include or exclude any com- 
modity ivhose price has varied inversely (Proposition XXXVI.), 
since this commodity retains constant exchange-value, not in 
all things, but in all commodities, and so is constant in exchange- 
value in all things other to money, and the addition or sub- 
traction of such a commodity to or from the standard by which 
the variation of money is measured does not alter the relation 
of that standard to money." Thus the standard of all com- 

■^ Here if r = 1, indicating constancy, both tliese Propositions fall directly under 
Proposition XXXII. 



THE POSSIBILITY OF CONSTANCY 51 

raodities (excluding money) may take the place of the standard 
of all things (including money). 

§5. We may also perceive the following relations, provided 
we suppose that the things be the same (or similar and in equal 
quantities) at all the periods compared. Then, v^hen any one 
ihing has remained constant through compensatory variations of 
others, all the others collectively have remained constant (Proposi- 
tion XXXVII.). For as the changes in the other things have 
compensated the relations of this one thing to all other things, 
they have compensated the relations of all the other things to 
this one thing, and as this one thing is constant, they have 
compensated their relations to something constant, and there- 
fore themselves, collectively, are constant. Also, according to 
Proposition XVIII. , if the one thiug had risen or fallen in 
exchange-value in every other thing alike, it would have de- 
pressed or raised all the others individually somewhat (and 
therefore collectively as many times more) ; therefore if it 
neither rises nor falls in every other thing, it cannot influence 
them either to a rise or to a fall, and they, individually (and 
collectively), must remain constant. Here, however, we are 
supposing the one thing to change relatively to the others in- 
dividually, but not collectively ; yet it is evident that, although 
the others change relatively to it individually, they cannot do 
so collectively. It follows that cdl things taken toe/ether, namely 
cdl things permanent over cdl the periods compared, must he con- 
stant in general exchange-value when any one is (Proposition 
XXXVIII.). And even if no one thing has remained con- 
stant, yet as any one by rising depresses all the others, and 
each of the others less in proportion to their numbers (accord- 
ing to Proposition XIX.), and reversely by falling, it is plain 
that all things collectively, provided, they be the same (or similar 
and in equal quantities) at cdl the periods compared, must be con- 
stant in general exchange-value (Proposition XXXIX.). This 
principle will call for further elucidation later, when the need 
of the proviso will be proved, and it will be shown in which 
kind of general exchange-value, when there are variations, the 
proposition is true. 



52 THE COERELATION OF EXCHANGE-VALUES 

§ 6. To return to the earlier principles about the possibility 
of compensation, it follows as a resume of them that it is pos- 
sible for the general exchange-values of two or any greater number 
of things up to all but one to vary, and yet for the general ex- 
change-values of all the others, that is, of any number of things 
from all but tioo doion to one, to remain constant (Proposition 
XL.). Of course, for this result to happen, the variations must 
be in certain proportions, which it is our object to discover. As 
yet we only know the general possibility — and not even that 
this possibility must necessarily always exist, but that it may 
exist. We know also an impossibility, which is, that if any 
one thing alone rises or falls, it is impossible for anything to 
retain the same general exchange-value, since the counter- 
Aveight is lacking. 

All this might seem to apply also to the case of general pur- 
chasing power. It can be so if this term be identified with 
exchange-value in all other things. But the term has not 
always been used in this, its proper, sense ; and then what is 
here proved of general exchange-value cannot be proved of 
what is meant by " general purchasing power." This subject 
will be discussed later. 

The statement that one thing is remaining constant in gen- 
eral exchange-value while other things are varying in their 
general exchange-values, means also, as we have seen, that this 
thing's particular exchange-values in those varying things are 
varying and are compensating for one another. This statement 
may be true of money — of which, in fact, we proved it ; — and 
as the particular exchange-values of money are inversely meas- 
ured by prices, it follows that a change of price, or many 
changes of prices in one direction, may be compensated by one 
or many changes of prices in the opposite direction, with the 
resultant that money remains constant in general exchange- 
value, the changes of prices neutralizing one another and yield- 
ing the same combined injfluence as if none of them had taken 
place. Therefore, although we have seen that a change of a 
single price cannot directly indicate the extent of the thing's 
variation in exchange-value in all things, it is possible for a 



THE POSSIBILITY OF CONSTANCY 53 

change of one price to do so if there is at least one other change 
of price in the opposite direction to the proper extent for coun- 
terbalancing ; or a change of any prices may do so, provided 
they are all together such as to leave the exchange-value of 
money unchanged. And although we have seen that a change 
of all prices alike does not indicate an exactly inverse change 
of money in exchange- value in all things, yet such a change of 
all prices but one may do so if that one changes in the same 
direction still more to the proper extent for counterbalancing 
in all others the opposite change of money in them. We are, 
however, more interested in the former case of money being 
stable in general exchange-value, which permits particular 
changes of prices directly to indicate the true variation in ex- 
change-value in all things, or in all commodities, of the things 
whose prices change. This can be, if all the changes of prices 
counterbalance one another, in pairs or otherwise. We shall 
later see that such counterbalancing is not wholly left to acci- 
dent — which if it were, the chances might be as small of its 
ever happening as of the proverbial letters falling out of a bag 
and composing the Iliad. There is no connection between 
those letters : there is close connection between all exchange- 
values. 



CHAPTER III. 

ON THE MEASUKEMENT OF GENEKAL EXCHANGE -VALUE. 

I. 

§ 1. The correlations reviewed in the preceding chapter — the 
variation of other things in exchange- value consequent upon a 
variation of any one, and the need of compensation in the varia- 
tions of other things in order to keep any one thing constant — 
seem to point to a peculiarity in exchange-value such as to 
separate it off from other quantities that are subjects of mensu- 
ration. In other subjects of mensuration we consider ourselves 
justified in thinking we have some things possessing fixed 
quantities of the attribute to be measured, by comparison with 
which we can measure the constancy or variation of the things 
of which the constancy is not known or the variation is suspected. 
Thus in measuring lengths at different periods of time we think 
we can take the circumference of the earth as a fixed and un- 
changing length, or even as such the length of any carefully pre- 
served stick at the same temperature, or, more abstrusely, the 
length of a pendulum swinging in a second at a given spot. But 
in the measurement of exchange-value at different periods we find 
that not even all other things always possess a fixed exchange- 
value, much less any selected one from amongst them. It is 
as if we lived in a world of gases and were seeking to measure 
length. In such a world, in Mdiich a change in size of one 
thing would involve a change of all others, perhaps we might 
succeed in the attempt, perhaps not. In regard to exchange- 
value, however, the case is not hopeless. In fact, the way out 
of the difficulty has already been indicated. The other things, 
taken collectively, are constant in certain circumstances, and in 

54 



COMPARISON WITH OTHER MEASUREMENTS OO 

others their collective variation will be measurable when the 
variation in them of the one thing in question is measured. 

The most striking difference is this. In other matters we 
think we have some fixed quantity of the attribute to be meas- 
ured, in small and convenient compass, permanently held for 
us by some material thing, which fixed quantity of the attribute 
in question can serve as a small unit whereby we may measure 
other larger or smaller quantities of the same attribute in other 
things. In matters of exchange we find not even the appear- 
ance of any fixed quantity of exchange- value being permanently 
held for us by any material thing, so that we are not justified 
even in assuming any fixed quantity to be attached to any ma- 
terial thing we know of; or, in other words, we have no natur- 
ally provided material on which Ave can lay out a small unit of 
exchange- value serviceable for measuring at different periods 
(or places) other exchange-values larger or smaller than it- 
Thus other measurable things at least seem to admit of our form- 
ing absolutely fixed small units, by comparison with which we 
seem to be able to speak of the constancy or variation of abso- 
lute quantities of the attribute possessed by other things. And 
consequently people have come to think of our possessing abso- 
lute measures in those subjects. But in exchanges we have no 
even apparently fixed small unit, so that we seem to be without 
power of measuring the variations of anything in absolute ex- 
change-value. Exchange- values are recognized as only rela- 
tive quantities ; or exchange- value is viewed as only relatively 
quantitative. And so exchange-value has been distinguished 
as merely relative from other quantities, in which it is sup- 
posed we have something absolute. 

§ 2. Hence it has been the burden of the song of many econo- 
mists, who sometimes appear to find pleasure in the thought, 
that the measurement of general exchange-value through the 
course of time or in distant places is impossible because nature 
has not provided us with any material permanently and every- 
where possessing the same general exchange-value. That 
nature has not rendered us this service is a fact, although 
we have seen error in the view that this omission is due to 



1 



56 MEASUREMENT OF GENERAL EXCHANGE- VALUE 



the essential impossibility of anything having fixed exchange- 
value. Others, again, have despaired of measuring general 
exchange-value on the ground, which we have seen to be in- 
sufficient as well as incorrect, that it is not an attribute in 
things like weight or color. Still others have denied that we 
can measure exchange-value at all, for the reason that it is a 
relation, and that we cannot measure relations.^ None of these 
is a good reason for despairing. 

The only real difference so far advanced is in the mistaken 
attitude which people have adopted. In other subjects of men- 
suration we have nothing more absolute than what we have in 
the mensuration of exchange-value itself. The differences 
which exist are differences of degree, and not of kind. 

II. 

§ 1. That exchange- value is a relative quantity, is not a 
peculiarity at all. All quantities are relatively quantitative. 
When a branch two feet long grows to be three feet long, it has 
changed from twice to thrice the length of a foot-stick. But 

^ F. A. Walker : " Value is a relation. Eelatious may be expressed, but not 
measured. You cannot measure the relation of a mile to a furlong ; you express 
it as 8 to 1," Money, 1877, p. 288 (repeated in Jloney, Trade and Industry, 1879, 
pp. 30, 60). The last statement is merely like saying that we cannot measure, 
but only express, the relation of a dollar to a dime. But this does not prevent us 
from measuring other exchange-values by dollars or dimes just as Ave measure 
other distances by miles or furlongs. — Macleod : "An invariable standard of 
value . . . is in itself absolutely impossible by the very nature of things. Because 
value is a ratio : and a single cxuantity cannot be the measure of a ratio. A 
measure of length or capacity is a single quantity : and can measure other single 
quantities. . . . But value is a ratio, or a relation : and it is utterly impossible 
in the very nature of things that a single quantity can measure a ratio, or a rela- 
tion. It is impossible to say that a : h : : x," Theory of credit. Vol. I., p. 213. 
Length is no more, and no less, a single quantity than exchange- value. As ex- 
change-value involves a relation between things in exchange, so length involves a 
relation between points in space. Neither is a mere relation, the one being the dis- 
tance between two points, the other a power in one thing of purchasing another. 
Measurement is only of quantitative relations. In measuring length we say that 
the length of the thing in question is to afoot (the length of certain sticks) as a; : 1. 
In measuring exchange-value we say that the exchange-value of the thing in 
question is to a dollar (the exchange-value of certain bits of metal) as a; : 1. The 
processes are exactly alike. W^e measure one relation by another relation. — 
Laughlin : " Since value is a relation there never can be an absolutely perfect 
measure of it. . . . Length is positive, and not a relative thing," op. cit., p. 100. 
In reality it is only because value, and length also, are relative (quantitative), 
that they can be measured at all. 



THE STAxVDARD IN SIMPLE MENSURATION 57 

the foot-stick itself has changed from a half to a third of the 
length of the branch. Only here we say, it is the branch, and 
not the foot-stick, that has changed in length. We do so, 
as already remarked, because we have other things in mind, 
and the foot-stick is the one Avhich keeps its length unchanged 
relatively to them. Of course it cannot be supposed that we 
know it is the branch which grows because it is living, and 
the foot-stick is constant because it is dead, wood. For we 
must know which sticks grow and which are constant, before 
we can know the distinction between living and dead wood. 
And we are not endowed with instinctive or innate knowledge 
as to which it is among things which change relatively to one 
another that is constant and which it is that is varying. We 
acquire this knowledge only by comparison with many things, 
— and comparison is the primary act in measuring. Thus even 
the mensuration of length is a relatively quantitative affair. 
We measure the foot-stick by other things before we pronounce 
it a good measure for measuring other things.^ There is, then, 
some standard in this mensuration superior to the material 
instrument we use as our measure. And so in other subjects 
of mensuration. What is the nature of this standard ? 

§2. In measuring some attributes or qualities, we employ a 
standard foreign to them, which we assume to be a fixed stand- 
ard, so far as they are concerned. We do so because any vari- 
ation that might occur in it would be independent of variations 
in them, and would be the subject of another measurement. 
Therefore the variations in them, measured in this standard, 
are regarded as absolute variations, over against their variations 
in comparison with one another, as relative variations. Yet 
this view of the absoluteness of the variations measured by this 
standard, does depend upon the further question as to M'hether 
this standard can be found to be really fixed, and so leads us 
back to a further measurement. Thus in the general subject 
of value, the cost-value of a thing is to be measured by the 
quantity of labor required to produce the thing, and itsesteem- 

^ T. Mannequin : " We do not measure length in the meter, nor capacity in the 
liter," Le prohllme monetaire, 1879, p. 28. Of course while using a meter-stick 
we do not stop to measure it. But we confide in the results of previous measure- 
ments. 



58 MEASUREMENT OF GENERAL EXCHANGE-VALUE 

value by the esteem in which it is held, which again is to be 
measured by the quantity of the thing forthcoming relatively to 
the intensity and the extension of human desire for it. There- 
fore, no matter how much a thing's exchange-value in all other 
things may change, its cost-value or its esteem-value does not 
change unless its cost of production or its rarity changes, and 
if these do not, it may be looked upon as absolutely constant. 
But now the question is renewed as to whether cost of produc- 
tion, or quantity of labor, can be absolutely measured, — which 
involves questions about both absolute time and absolute feeling 
of effort ; — and as to whether quantities of matter, or masses, 
can be absolutely measured. In all cases we get back to a 
measurement of an attribute or quality that can be measured 
only by some other instance of the same attribute or quality, 
which leads us to the question how this is to be measured, and 
again and again to the same question, wherefore we want some- 
thing ultimate. These attributes are generally the first ones 
we start with the attempt to measure, such as length, weight, 
and a few others. Among them is general exchange-value, 
although this has been late in calling for measurement. They 
may be called subjects of simple mensuration. 

§ 3. In subjects of simple mensuration we have nothing, in 
the sense of some material thing, or individual instance of the 
attribute measured, that is ultimate or absolute — nothing that 
w^e can know, or even conceive, to be constant by itself alone, 
without relation to something else. The only ultimate standard 
we have here is, in fact, itself a relation. It is the relation be- 
tween the whole of the quantities to be measured and the 
quantities themselves as its parts. And the only constancy or 
fixity we can have is the constancy or fixity of some part rel- 
atively to this whole. We see this most clearly in that most 
perfect of all mensurations, the mensuration of angles. Here 
the unit is not really the degree, but the whole circle, of which 
the degree is only a fixed part. In measuring an angle, we 
compare it with the whole circle of which it is a part. And 
an angle is fixed, or constant, as it keeps the same relation to 
this whole. And so in measuring the length of anything, we 



THE STANDARD IN SIMPLE MENSURATION 59 

are really comparing it with a larger whole — if uot the whole 
extended universe, the whole earth, or at least the whole of 
things we see around us. Thus when a branch and a foot- 
stick change in length relatively to each other, we perceive that 
the branch changes in length relatively to the earth, and we 
do not perceive any change in the foot-stick relatively to the 
earth. To be sure, the foot-stick has changed relatively to one 
other thing. But we know of other things becoming smaller 
at the same time, in comparison with which the foot-stick has 
become larger, and, among the infinitude of things with infin- 
itely different lengths, we are unable to perceive any change in 
the length of the foot-stick (at the same temperature) compared 
with the whole of things. If we did (as we do, in fact, at 
different temperatures), we should have to say that the foot- 
stick had changed in length. 

§ 4. The fixity is not necessarily in the whole itself inde- 
pendently of its parts, nor in the parts independently of the 
whole. The only fixity we can have knowledge of is in the 
relation between the parts and the whole. 

It is not necessarily in the whole itself; for our universe 
might expand or contract in size without our knowing it, so 
long as all particular things, including our bodies, kept the 
same relationship to the whole, as we perceive it, — and who 
knows, or who cares, whether the universe of material things is 
one day half as large as another in so-called absolute size ? 
Such a change would, after all, have to be a change in relation 
to something else than our material universe — say, another 
material universe, or something called absolute space. Then 
if such a cliange actually did take place, what we should desire 
for our measure of length would be, not absolute permanence — 
permanence in relation with that extra-mundane thing, — but 
relative permanence, in relation to our own universe or world. 
What happens to our universe and its parts in comparison with 
something else beside it, in no wise concerns us, and we do not 
care to measure this relation. Moreover, even if we did find 
such a change, we could not think of it as being more in our 
universe than in that other thing, except by comparison with 
still another thing ; and so on without end. 



60 MEASUREMENT OF GENERAL EXCHANGE-VALUE 

Nor is the fixity necessarily in the parts. In one physical 
theory our material universe is conceived to be composed of ex- 
tended atoms, which do not touch one another. Now suppose 
that all atoms are approaching toward or receding from one 
another, that is, that the distances between the atoms are en- 
larging or decreasing compared with the distances through the 
atoms. Such a change, say of contraction, may be stated in 
two (among five) typical ways : the one, that the atoms are per- 
manent in size, and the intervening spaces are decreasing ; the 
other, that the intervening spaces (between the centers) are per- 
manent, and the atoms are enlarging — in each case relatively 
to something else outside our universe. By the physicist the 
former of these interpretations is the more likely to be adopted, 
because it fits in better with his preconceptions, which generally 
do not take into account what is meant by permanence in size 
— namely, permanence in relation to some other size. Yet, if 
we could actually see this change going on, we should be as 
likely to say that the atoms are growing larger as to say that 
the universe is growing smaller. Practically, however, so long 
as we remain out of touch with these smallest things, if the 
things which are the smallest for us, and all the things which 
we see and feel, including our bodies, grew smaller compared 
with the atoms in them in exactly the same proportion (or on 
the average, allowing for relative changes in some things, such 
as go on anyhow), so that they all (with these exceptions) re- 
tained the same relations to the whole, we should want our 
measure of length, at least for all practical purposes, to decrease 
with the rest, so as to retain the same relation to the whole 
as it had before, the sameness of this relation being our stand- 
ard, although then our measure of length would decrease in 
size relatively to the atoms, which are, according to the hy- 
pothesis, outside our visible and tangible world. Then, prac- 
tically for us, measured by the standard which we use, it would 
be the atoms that are growing in size.^ 

§ 5. In a similar manner, when one thing, say a branch, 

2 Probably the measurement of length by the pendulum would indicate a 
change. But then we would be at a loss as to whether the change were in our 
usual mensuration of length, or in our mensuration of time. 



THE STANDAND IN SIMPLE MENSURATION 61 

changes in length before our eyes in comparison with other 
things, it would be possible to assume that the branch has re- 
tained the same size and all the rest of our universe has grown 
smaller, — that is, relatively to something else beside the uni- 
verse. But if we adopted this view, — and if it were true, — 
we should still want our measure of length to go with the rest 
of our universe, instead of remaining constant with this one 
thing in its mystical connection with something else about which 
we know nothing beyond its agreement with this thing. We, 
therefore, adopt the opposite course, and think, not really of 
our universe as unchanging, but of the unchanging measure of 
length as the one which remains permanent in relation to our 
universe, and as the branch is changing in relation to our uni- 
verse and to tliis measure, we think of the branch as changing 
in length. It is not that we bring in any theory of probabilities, 
and argue that it is more probable that the branch changes and 
that the universe does not. What we see is that the branch 
does change relatively to the whole, and to other things which 
are not seen to change relatively to the whole. While it is 
doing so, the universe and many things in it do change equally 
much relatively to it, but we do not see that the universe does 
change at all relatively to the things, the foot-sticks, which we 
have chosen for our measures. Hence, we assign the change 
to the branch, and not to the foot-stick or to the universe. 
What we measure is a fact, not a probability. 

What is here said of length, is true also of other subjects of 
simple mensuration. So, for instance, in the case of mass of 
matter. A body has permanent mass, if it always has the 
same quantity in relation to the whole quantity in our universe. 
This is our ultimate conception of permanence of mass ; for we 
know nothing about any other permanence of the mass of the 
whole universe itself, or of any of its parts, which perma- 
nence, if it existed, would have to be in relation to something 
else. If new matter were injected into all things in the same 
proportion, we might not perceive any change, and if some- 
how we did learn of it, we should be glad that our measures of 
mass were increased along with the rest. Similarly in the case 
of force, in its simplest forms. 



62 MEASUREMENT OF GENERAL EXCHANGE-VALUE 

But if any of these hypothetical and miraculous changes 
took place in a part, say a half, of our world, especially if 
scattered about, we should be at a loss where to place it 
(whether to conceive of it as an increase in the one half, or as 
a decrease in the other), and should distribute it over the whole ; 
and then our measure would be some size, or some weight, 
which has the same relation to the whole as our old measures 
had. 

§ 6. In all these subjects we are somewhat careless of the 
widest standard theoretically possible, and content ourselves 
with the most practicable and the most prominent. Thus our 
ordinary measures are measured, not by their relation to the 
whole material universe, but by their relation to the earth, or 
even to particular regions on the eartli, and have even been 
measured with more especial reference to the sizes of our bodies. 
If, therefore, such changes as just supposed were to take place 
in the outer stellar regions, we should probably pay no atten- 
tion to them as regards our measures of length or of mass ; 
for the relation between our measures and our standard for 
them, the earth, would be unchanged.^ Again, if something 
were miraculously increased on our earth, we should probably 
think it had received matter from the outside, and not alter 
our conceptions. But if we all woke up some fine morniDg 
and found half the things with which we are familiar become 
larger compared with the other half, and this other half there- 
fore smaller compared with those, then, even if all our meas- 
uring sticks happened to remain alike and were constant with 
the one or the other of these sets, we should have to adapt our 
measure of length to some average of all these changes ; for 
to say that our measuring sticks had remained constant, would 
be to take the half of things with which they continue to 
agree as the constant ones, although we should have no more 

^ At all events in our measurements of things on the earth. For in astronomy 
we have to make use of other bodies beside the earth in our measurement of dis- 
tance. It is conceivable that the astronomical and the geographical miles could 
vary. If the world were still molten and were contracting, we being salamanders 
that live in fire, we might find this to be the case. Perhaps we could find it even 
now if our measurements were suificiently accurate. 



THE STANDARD IN SIMPLE MENSURATION 63 

right to do this than to take the other half as the constant 
ones. What we should desire for our standard of length would 
be some portion of the lengthened things, and some addition 
to the shortened things, that has the same relation to the whole 
as our standard measures had before. 

§ 7. To be sure, such variations are not found to take place 
in the lengths and in the weights of things on the face of the 
earth, — which is why we should regard them as miraculous if 
such changes should suddenly begin to take place. When we 
open our eyes every day, what we see contains many things — 
mountains, plains, rocks, buildings and land-marks of thousands 
of kinds and descriptions, — that do not appear to alter, do ap- 
pear to remain the same, in their relative positions and in their 
relative distances or lengths. And many things, when they are 
left to themselves, appear, when we weigh them, to have the 
same weight (relatively to many other similar things) always. 
Upon these we hit for our standard. Thus the foot-stick, or 
the pound weight, which we use for measuring, is one which 
we find not to vary in relation to other things — to be one 
among the things that do not apparently vary relatively to one 
another. Yet, although this constancy of so many things rela- 
tively to one another obscures, it does not alter, the principle 
of mensuration. It is only because the foot-stick (at the same 
temperature) is not j)erceived to vary relatively to the whole of 
things, that we are justified in thinking of it as a constant 
measure. We are not justified in thinking so simply because 
there are some things which keep constant relations of length 
amongst themselves. There must be enough of these to form 
a whole, in which not only our measures, but all the things in 
which we are interested are included. And in this whole must 
be included also the things which do vary. And if these things 
which do vary are not to affect the measure by its variations 
relatively to them, it must be because of compensations in their 
variations. For instance, if all the things that vary in length 
varied only by growing larger relatively to the things that are 
constant amongst themselves, which then are always becoming 
smaller relatively to those other things, then (supposing those 



64 MEASUEEMENT OF GEXEEAL EXCHAXGE-VALUE 

other things to be sufficiently important to interest us) we should 
not consider the things stationary amongst themselves to be a 
good standard by which to measure the others. We should 
want to include the others in the whole, and the measure would 
have to increase partly with the increase of the others. We are 
saved from this need only by the fact that the things which do 
vary in length relatively to the things stationary amongst them- 
selves, become both larger and smaller in about the same pro- 
portions. Trees which grow, also decay and fall or are chopped 
to bits, and disappear, as others appear. Animals increase, and 
die away. Clouds form, and fade. Heat expands, and cold 
contracts. It is only because our foot-sticks, which remain 
steady in relation to the things that do not change amongst 
themselves, remain steady also in relation to all these increasing 
and decreasing things, on the whole, to all appearance, that 
they are suitable measures of length. 

§ 8. Now the exchange- value of everything forms part, with 
the exchange- value of everything else, of a whole. By com- 
parison with this whole we ought to be able to reach the same 
kind of fixity in this subject as in other quantitative matters. 
To be sure, we are here left without the help of things that re- 
main constant relatively to one another. In the economic 
world there are no mountains, plains, rocks, buildings and other 
land-marks that retain the same exchange-values relatively to 
one another. We have only things that change relatively to 
one another. Yet this difference is only a difference of degree, 
and not of kind. The relative fixity in length of land-marks 
inter se is a help, and a great help, but not a requirement, in 
the mensuration of length. The standard is the relation of a 
part to the whole — or to a practicable whole. This standard 
we can have in the measurement of exchange-value, as well as 
of other attributes. The principle of simple mensuration is the 
same in all cases. In measuring exchange-value what we need 
is to form a proper conception of the whole of exchange- values. 
This done, we shall be able to compare with it the exchange- 
value of anything, in order to find whether this exchange-value 
is constant or varying in relation to the whole, and how much 



THE STANDARD IN SIMPLE MENSURATIOX 65 

varying. If it be constant, not only the whole including it, 
but the whole of all the rest of things, will be constant rela- 
tively to it. If it rise, the latter whole will fall ; and reversely ; 
diverging from the fall or rise of the whole including the thing 
in a, definite proportion. It will, then, be sufficient to be able 
to compare the exchange-value of any one thing with the ex- 
change-value of all the others. 

If we succeed in this, the absence of any particular thing 
that naturally keeps its relation to the whole permanently un- 
changed will in no wise prevent us from reaching satisfactory 
results. We shall have a method of measuring exchange-value, 
in the place of a measuring instrument. Instead of having a 
bit of material which keeps the unit tolerably constant, we 
shall be able to lay out a unit of exchange-value upon materials 
that vary in exchange-value. In doing so, however, we shall 
do only what is done in all other matters of mensuration when 
the greatest possible accuracy is desired. The metal used as 
bearer of the unit of leng-th does not remain constant under 
changes of temperature, and the engineer must allow for its 
variation, often using as a foot a length which is not the length 
of his foot-stick. But more than this. The possibility exists 
that, although nature does not provide us with a material un- 
varying in exchange-value, we shall be able to make such a 
thing for ourselves, with tolerable exactness, and so be in pos- 
session of a dollar-bit of metal or paper comparable with a foot- 
stick of wood or metal. 

That exchange-value is not something absolute, is, therefore, 
no objection against our being able to measure it with as great 
precision as we can attain to in other subjects of mensuration, 
since in no subject of mensuration is the attribute itself, which 
we measure, anything absolute.^ And what we have of fixity, 

* J. Gamier: " Unfortunately, all value being essentially variable, it follows 
that there cannot be an invariable unit of value, and that we cannot estimate the 
absolute magnitude of the value of things, but only their relative and comparative 
magnitude. . . . The value of this sum of money [by which we attempt to meas- 
ure the value of a house] is not value existing by itself, abstraction being made of 
all comparison, and we can form an idea of it only by comparing it with all the 
things we can obtain in exchange for it [including the house itself] .... It fol- 
lows therefore from the internal nature of value that the search after a (mathe- 

5 



QQ MEASUREMENT OF GENERAL EXCHANGE- VALUE 

or of absoluteness in this sense, in other measurements, we have 
also in the case of exchange-value, so far as its relativity is con- 
cerned — namely, fixity of the relation between a whole and its 
parts. ^ 

§ 9. It is strange how great is the fondness for absoluteness, 
and how great the dislike of relativity. In subjects wholly 
relative, where much was once carelessly thought to be absolute, 
people are fain to retain the absolute in some nook or corner, 
letting it in by the back-door after driving it out in front. 
The eminent mathematician and economist, Cournot, drew the 
distinction that we cannot have an absolute exchange-value, 
but we can have an absolute variation of exchange-value.^ 

Such a distinction is due to an ambiguity in the term " abso- 
lute." The literal meaning of " absolute " is " without relation 
to anything else." It has acquired the meaning of " really fixed 
or permanent," or " really invariable in its relations." Besides 
which, the adverbial form is often used in a merely intensive 
sense, as when we say " absolutely all," meaning " all without 
any exception whatsoever." Plainly it is only the second, the 
acquired, sense that has any importance in matters of mensura- 
tion. Nothing — not even the universe, or "absolutely" all 
things — can be absolutely variable or absolutely constant in 
size or in any other quantitative attribute, in the original sense. 

matically) exact standard or meter of value is impossible," op. cit., p. 290 (the 
italics in the original). This passage finds fault with the mensuration of value 
for being able to be no more than it ought to be, a measurement of the relative or 
comparative magnitude of values. It also makes the mistake of implying that 
the meter, the unit of length, is not only a mathematically exact, but also an ab- 
solute, standard, existing by itself, with abstraction of all comparison. Thus the 
fault found with the mensuration of exchange-value is that it is not what the 
mensuration of length also is not. 

5 We have seen that esteem-value is measured by esteem, and cost-value by 
labor-cost. AVe may notice, in passing, that esteem and cost are to be measured 
in the same way as above described. We have a certain esteem for all the things 
we possess. If the things we possess become more numerous, the size of each com- 
pared with the whole decreases. So our esteem for each decreases. And with it 
each thing's esteem-value. Reversely if our possessions grow less. Then every 
single one becomes larger in relation to the whole ; consequently it grows in 
esteem, and in esteem-value. Similarly with cost-value. If an hour's work 
comes to produce more things, each of these becomes smaller in relation to the 
whole product : it falls in labor-cost, and in cost-value. 

6 Op. cit., p. 22. 



TWO GENERAL STANDARDS 67 

But, in the acquired sense, a thing can be absolutely permanent 
as well as absolutely variable in relation to the standard 
adopted — and if a thing be found absolutely permanent, say in 
size, its size might just as well be spoken of as absolute — in this 
secondary sense. Exactly so with exchange-value. Cournot 
himself distinguished "absolute variation " from " relative vari- 
ation " only because when two things change in exchange-value 
relatively to each other, we can conceive of the change as being 
wholly in the one." He apparently failed to see that in this 
case we are merely comparing each of the things with all things, 
and the change is wholly in the one only because this one is 
changing relatively to all other things, while the other is not 
changing relatively to all other things. The variation of the 
one, then, is as relative as the permanence of the other. And 
if the variation of the one can be called absolute, because of the 
comparison with the true standard, so can the permanence of 
the other. And if we can say a thing can be absolutely per- 
manent in exchange-value, we can equally well say it is perma- 
nent in absolute exchange-value. All this use of the term 
" absolute," however, should be avoided when dealing with 
quantities, quantities being wholly relative. For to use it in 
the acquired sense involves the risk of importing into it also 
the literal sense of " without relation." 

III. 

§ 1. Even the distinction between two kinds of general ex- 
change-value — its division into exchange-value in all other 
things and exchange-value in all things — is not peculiar to our 
subject. Thus in our extended universe if something in chang- 
ing in its size does not aifect the size of the whole, this distinc- 
tion exists ; for if, for example, the thing were reduced to half 
its size in relation to the whole, it would be slightly more than 
halved in relation to the other things. Or again, if in chang- 
ing in its size it affects the size of the whole by so much, then, 
if for example it were halved compared with what the whole 
used to be, it would be halved compared with all other things, 

' Op. cit., p. 18. 



68 MEASUEEMENT OF GENERAL EXCHANGE-VALUE 

but not quite halved compared with all things. The distinction 
may be more plainly seen in the case of motion, which yields a 
closer analogy with our subject — already used in a partial way. 
In a finitely extended universe of points, the whole of which 
we cannot conceive to move, since there is nothing in relation 
to which it could move, if all others of its points retain their 
positions unchanged relatively to one another, the apparent 
motion of a single point a certain distance (compared, say, with 
the diameter of the whole) will be its real motion. This is its 
motion relatively to all other points. Again, in a finitely ex- 
tended material universe, in which its parts have attractions so 
that for the whole there is a center of gravity, which we can- 
not conceive to move (except in relation to other parts of the 
universe), since its motion would represent the combined mo- 
tion of the whole, and we cannot conceive of the whole as mov- 
ing, for the same reason as before; then, all other bodies retaining 
their positions unchanged relatively to one another, the motion of 
a single body a certain distance through the others in a certain di- 
rection would displace the center of gravity in the whole a slight 
distance in the same direction, and therefore, this center being 
conceived as unmoved, it would cause the rest of the universe to 
move a slight distance in the opposite direction (relatively to its 
center of gravity), and its own apparent motion (relatively to all 
the others) will not be its real motion (in relation to the center 
of gravity of the whole), which will be slightly less in the same 
direction. This is obviously its motion relatively to all things, 
including itself. Thus in the former case, or in the latter also 
if we conceive of the motion only as compared with all the 
other things, that motion is comparable with the rise or fall of 
the exchange-value of a single exchangeable thing in all other 
things ; and in the latter case, what we regard as the body's 
real motion is comparable with the rise or fall of the exchange- 
value of a single exchangeable thing in all things. In both the 
subjects, when our attention is called to the distinction, we 
regard the latter point of view as the truer one — the one alone 
suitable for all measurements, — but the former as also a pos- 
sible one. In the case of motion a body cannot move without 



TWO GENERAL STANDARDS 69 

pushing or pulling something else in the opposite direction, and 
if that something else retains the same position relatively to all 
other things, all these other things must have been slioved in 
the opposite direction, though only to a very slight extent, so 
that the motion of the body in question is not so far in reality 
as it is in relation to them, since they are moving slightly in 
the opposite direction. And so with the movement of an ex- 
changeable thing upward or downward in exchange- value rela- 
tively to all other things : it pushes them somewhat in the 
opposite direction, wherefore its motion relatively to them is 
greater than it really is, as they are moving in the opposite di- 
rection.' For convenience, however, especially as the opposite 
motion of all the other things is infinitesimally small (though 
less so in economics than in physics), we can, if we choose, re- 
gard all the other things, or even only those of them which re- 
main unchanged relatively to one another, as our standard, and 
measure not merely, as we do, the motion of a body by its re- 
lation only to all or even to some other things, but also the 
variation of a thing's exchange-value merely by its relation to 
the other things. 

§ 2. When we have chosen which method we shall adopt, 
and what shall be our standard, there is of course no occasion 
for employing in our measurements the law of probabilities — 
as was asserted also in this connection by Cournot.^ We do 
not say : it is more probable that all the other things have re- 
mained stationary than that this one has stood still and they 
moved ; or, it is more probable that all things have together 

1 If the moving body pushed something else equally heavy equally far in the 
opposite direction, or several things appropriately lesser distances, the motion of 
this body (no longer the only one moving compared with the rest) would be 
compensated, so that its apparent motion compared with all the common other 
things which have remained unchanged amongst themselves would be its real 
motion compared with all things (including itself), while those common other 
things would remain unchanged also in their real positions. The similarity with 
the case of exchange-value as exposited in Proposition XXXIII. is manifest. 
There is only one difference, which gives greater simplicity to the ease of exchange- 
value. The motion of bodies in space may be in three dimensions ; the motion of 
things in exchange-value can be only in one dimension — hence always only in 
either the same or opposite directions. In exchange-value there is no parallelo- 
gram of forces, excei)t as this is reduced to a straight line. 

- Ojy. cit., pp. 15-16. Similarly Bourguin, B. 132, p. 24. 



70 MEASUREMENT OF GENERAL EXCHANGE- VALUE 

remained stationary, wherefore both this and the others have 
moved relatively to the whole. But having adopted our point 
of view, we simply measure, as best we can, what we see hap- 
pening before us. And our point of view itself in these matters 
we adopt, not by any use of the law of probabilities, but because 
the myriad interrelations which do not change, or which do not 
change on the average, make more impression on us than the 
particular ones which do change. 

lY. 

§ 1. Having found these points of resemblance between the 
mensuration of exchange-value and the mensuration of other 
ultimate quantitative attributes, let us turn to a difference that 
is of considerable moment. 

The ultimate standard, consisting of a relation between the 
parts and the whole composed of them, would seem to demand, 
for its perfection, that the whole should be the same, or exactly 
similar, whole at both the periods, or at both the places, between 
which the comparison is instituted. This requirement is ob- 
served in the mensuration which we have already noticed as 
being the most perfect, namely the mensuration of angles. 
For here the circles with which we compare angles are so exactly 
similar that we do not hesitate to pronounce them the same, 
and even speak of all circles as being only one circle. And in 
physical matters the requirement seems to be satisfied ; for our 
physical world appears every day to be made up of things so 
exactly alike that we consider them to be the same things, and 
although there are some new formations and destructions of old 
things, yet a little induction teaches that the matter in these, or 
the ultimate bodies composing them, are constantly the same. 
But in economics the state of things is very different. From 
age to age, from century to century, even from year to year and 
from week to week, the economic w^orld is a different world, 
composed of many things at one time which do not exist at 
another. For our economic world is only a part of the whole 
material world, and may draw not only new bodies, but new 



THE TRUE PECULIARITY 71 

matter from it, and return old things to it. In the economic 
world there is creation and annihilation. If particular things, 
when consumed, were always replaced by similar things, and 
nothing new were produced, we should always have exactly 
similar worlds, which is all we want.^ But not only the par- 
ticular things appear in different quantities, constituting classes 
of different sizes, but wholly new classes come into existence at 
times, and some old ones pass out, or qualities become better or 
worse, really constituting different classes. And this is not all. 
At the same time the economic world in one locality is different 
from the economic world in another. 

Here we have what probably constitutes a difference in kind 
between the subjects of mensuration in economics and in the 
physical sciences. It may be that there is no creation or anni- 
hilation in the material universe, which therefore is the same 
whole always ; wherefore the economic world differs from that 
whole in kind. And this difference in kind would seem to go 
over into our mensuration of exchange-value compared with 
our mensuration of physical attributes, such as length and 
weight. 

§ 2. Yet compared with our practical measurement of phys- 
ical attributes this generic distinction does not exist in entirety ; 
for we never use the relation between the parts and the whole 
universe as our standard, but only the relation of the parts to 
some lesser whole within the larger whole, and this lesser whole, 
being only a part of the larger whole, may receive new mattei 
from the outside or yield up old matter to the outside. For 
instance, we measure the weight of bodies by their relation to 
the earth below them ; but through volcanoes the earth below 
them sends forth matter to the air above them, and from outer 
space it receives meteorites, so that the whole with which the 
weight of bodies is compared is a variable one. These changes 

^ For as it is not the other things themselves which constitute a thing's par- 
ticular exchange-values, but these are its power of purchasing the other things, 
it is indift'erent what the other things are, provided they be alike. The difficulty 
does not lie in the distinctness of the things, but in their dift'erences. The prin- 
ciple is broad. We do not know that the physical things we see every day are the 
same ; but tliis ignorance is no source of trouble to the physicist. 



72 MEASUREMENT OF GENERAL EXCHANGE- VALUE 

are relatively so small that their influence is imperceptible to 
us, and we neglect them. It is worth enquiring, however, 
what we should do if they were large enough to provoke at- 
tention. 

It is well known that the meter is supposed to be a definite 
portion of the circumference of the earth. Now if a small 
planet, or comet, were to collide with the earth and unite itself 
to it, perceptibly enlarging our world, and some of us should 
survive the catastrophe, it is probable that, as already remarked, 
we should regard the same meter to be, not the same proportion 
to the new earth, but 'the same proportion to that part of the 
new earth wliich alone constituted the old earth. Or if the 
planet in colliding with our earth should scoop away some of it, 
and carry it off into space, we should want our meter to remain 
the same proportionately to the smaller earth as it was before 
to this same part of the old earth. In other words, we should 
disregard accretions and subtractions, and use for our constant 
whole with which we compare the parts only a whole which 
exists at both periods compared — a whole common to both the 
periods. 

§ 3. In economic matters it is easy to imitate this procedure. 
There are even two ways of doing so. The one is this. In 
measuring the exchange- value of money at two periods when 
the material constitution of the economic world has varied, we 
might merely take, in every class of goods, the largest amount 
of it which exists at both periods. Thus if one class has grown 
larger, we should cut oif what has been added at the second 
period, and take into account only the quantity of the first 
period. Or if another class has diminished, we should disre- 
gard the surplus which existed at the first period, and take 
only the quantity of the second period. The world so reduced 
would be a world which exists at both the periods. And we 
might do the same if we were comparing two economic worlds 
at the same time, but in different places. 

The other way is this. Instead of taking what is common 
to both periods in every class separately, we might take what 
is common to both the periods in all the classes together as a 



THE TRUE PECULIARITY 73 

whole. Thus in measuring the exchange-value of money in 
commodities we should compare the total mass-quantities of 
goods a given sum of money Avill purchase at each period in 
the proportions in which the total sum of money is found to 
have been spent on the goods at each period. Or, reversely, 
in comparing the exchange-values, collectively, of all commodi- 
ties in money, we should compare the total sums of money a 
given mass-quantity of goods will command at each period, 
this mass-quantity being composed at each period of classes in 
the proportions in which the actual mass-quantity of goods is 
found to have been composed at each period. In order to carry 
out this method, we shall need to find what constitutes same- 
ness in a mass-quantity of goods at different periods, or in dif- 
ferent places. 

The second of these methods is conformable to the analogy 
of physical mensurations. In physical matters our standard 
whole does not have to be composed of classes of things indi- 
vidually the same or similar at both the periods compared. 
This is plainer in a more complex case. Suppose it should 
miraculously (as we should say) happen that some classes of 
physical things should be enlarged by creation of new bodies 
and others diminished by annihilation of old ones. Then we 
should want to eliminate only the excess in the total of the one 
period over the other. What we want in our standard whole 
is that it should at all periods be composed of the same amount 
of substance or material. Now in economic things the substance 
or material is utility — or the importance we attach to things. 
Hence, we want our whole at each period to contain the same 
amount of utility, or importance ; and it is indifferent how this 
is distributed in the various classes of things. 

It would seem, then, that even this difliculty, which threat- 
ened to be disastrous, may be overcome. The analogy of the 
mensuration of exchange- value with other kinds of mensura- 
tion may again be made perfect. The only peculiarity that re- 
mains in the mensuration of exchange-value is that here the 
wholes given us by nature are not the same at both periods, so 
that we have the task of reducing them ; while in physical 



74 MEASUREMENT OF GENERAL EXCHANGE-VALUE 

subjects the material world seems to be given to us by nature 
as a constant — or at least as a sufficiently close approximation 
to a constant. The economic world may be admitted to be dif- 
ferent in kind from the physical ; yet economic mensuration is 
the same as the physical. 

A certain defect must further exist in measuring the general 
exchange-value of anything. This is the impossibility of taking 
into account all the things which possess exchange-value even 
in ordinarily small economic worlds during even very short 
periods. There is a necessary confinement of our attention to 
the exchange-value of money in the most prominent classes of 
staple commodities. But, again, this defect is not peculiar to 
the mensuration of exchange-value ; for in all mensuration we 
omit notice of a major part of the universe, and generally our 
standards are relationships to lesser wholes within the complete 
whole. 

§ 4. It happens, however, that we do not need so much pre- 
cision in the mensuration of exchange-value as we do in the 
mensuration of many other quantitative attributes. We need 
precision in any measurement only to the extent necessary to 
prevent the discovery of misfits in any subsequent combinations. 
Thus if a surveyor, measuring round a field, reckons that his 
two last points are a certain distance apart, and then finds 
that they are not that distance apart, his first measurements 
have not been conducted with the accuracy desirable, and 
corrections must be made. But in the measurement of the ex- 
change-value of money over several periods there are few op- 
portunities for the exercise of correction more sure than the 
original calculation, if conducted on right principles and with 
care. At least this is the condition to-day ; for there is no 
knowing what degrees of accuracy may in future be obtain- 
able, and therefore be desired. All that we need at present to 
strive after is to find the proper method whereby w^e may 
rectify the carelessly made calculations, or guesses, which every- 
body is apt to make — some asserting that money has appreci- 
ated, much or little, others that it has depreciated, others again 
that it has not varied at all, — and to approximate as nearly as 



THE TRUE PECULIARITY 75 

possible to the truth, thereby reaching a result which nobody- 
can question.^ 

2 J. B. Say made the statement that in economics the problem of finding a 
constant exchange- value is like the problem, in geometry, of squaring the circle, 
in that both are unattainable with perfect exactness, op. cit., Vol. II., p. 89, cf. 
Coitrs comptetd' ccouomie jiolitique pratique, 2d ed., p. 181. This statement has 
often been repeated as if it expressed the hopelessness of all attempts to measure 
exchange- value. Yet if we could attain to anything like the approximation to 
exactness in measuring exchange-value between two periods — or in measuring 
anything else — that we can reach in measuring the ratio of the circumference to 
the diameter, we should have very good reason to congratulate ourselves. — Mac- 
leod has altered the simile by saying the search after an invariable standard of 
value is like the search after the philosopher's stone or perpetual motion, Ele- 
ments of political economy, 1858, p. 171. It is strange that what is not only a 
legitimate but a necessary problem in economic science should by an economist 
be likened to things which never were objects of science, but only of cupidity. 



CHAPTER lY. 

SELECTION AND ARRANGEMENT OF PARTICULAR 
EXCHANGE-VALUES. 



§ 1. In order to compare the general exchange- value of any- 
one thing (generally money) at diflf'erent periods, a preliminary 
labor is that of obtaining an expression for its general exchange- 
value at each period separately. To do this involves two dis- 
tinct operations. The one is to select and properly arrange 
the thing's particular exchange-values. The other is to com- 
bine these into the thing's one general exchange-value, which 
they compose. The latter has been the subject of more dispute 
than the former, which, though also a subject of many discord- 
ant opinions, has not received the attention it deserves, and is 
by no means so easy a problem as it has been taken for. It is 
the subject which naturally calls for attention first, although 
it may not admit of complete solution independently of the 
other. 

To obtain with complete theoretical exactness the exchange- 
value of money — or, to be precise, of a certain sum of money, 
say the money-unit — at any place during any period — a week, 
a month, a year, — we ought to take account of every individual 
thing which has been exchanged at that place during that 
period, and of its price or exchange-value in money, which will 
give the money-unit's exchange- value in it, whether it was 
actually exchanged for money or not. To do this is impossible ; 
and so our practical measurement of general exchange-value 
during any period is subject to curtailment. The whole which 
we can employ can only be a part of the total of exchangeable 

76 



EXCLUSIONS AND INCLUSIONS 77 

things. But our eiforts must be directed at making this prac- 
ticable whole as large as possible ; for, as has been said by an 
eminent investigator in this subject, " the result is more accu- 
rate, the greater the number of the data, and the smaller the 
number of omitted articles." ^ 

§ 2. The curtailment must begin by leaving out of account 
things which appear only as individuals — such as race horses, 
paintings, antiques and the like. These individual things must 
be omitted not only because their number is legion, but because 
each one is exchanged only occasionally, and so would not ap- 
pear in all the successive periods, and none can stand for an- 
other. Moreover their omission is only a small loss, as, in spite 
of their great numbers, their sales all told form but a small part 
of the immense quantity of all sales. 

The majority of exchangeable things fall into classes, under 
generic or specific names, in which all individuals at the same 
time and place have the same price, or different prices accord- 
ing to different qualities, which form sub-classes ; and there is 
always a succession of similar individuals appearing in the 
market during every period. These classes we can employ in- 
stead of the individual things themselves, and so our labor is 
already enormously reduced. For when one of the classes, 
representing the individuals in it, is said to vary in exchange- 
value or in price, we at once know that all the individuals in it 
have so varied. The quoted price of a bushel of wheat is not 
the price of a particular bushel of wheat, bnt of any bushel of 
wheat (of the same quality) at the same time and place. 

All classes of things, however, do not equally well represent 
the individuals in them ; for in some classes the individuals 
vary infinitely in quality or in size or in many attributes that 
go to make them valuable. Thus all complex products, such as 
machines, buildings, ships, railroads, are too variegated in their 
individuals to make the logical subsumption of these into classes 
of much importance for the object we have in view. Therefore 
these things, as also fancy breeds of animals, precious stones, 
and most articles of luxury, must be neglected as being little 

1 Edgeworth, B. 60, p. 197. 



78 SELECTION OF PARTICULAR EXCHANGE-VALUES 

more than collections of iintractable individuals. The omission 
of complex products is of little consequence, because their ex- 
change-values generally vary in somewhat the same way as do 
the materials of which they are made or the simple products 
which they help to make. Land also is such a heterogeneous 
class, embracing lots and fields and forest districts of infinitely 
various exchange- values according to situation and natural fer- 
tility. The omission of these from an attempt to measure the 
general exchange-value of money in all exchangeable things is 
of considerable moment. But if we are seeking rather to es- 
tablish for the exchange-value of money a standard composed 
of products, land would not belong to this. StocTcs or shares 
in railroads and industrial companies are not to be counted, 
partly because the prices, or money costs, of these are not to be 
counted for the reason just given, and partly because their 
prices are dependent upon the profits, which are greatly depend- 
ent upon the prices of the services or products, already counted. 
For a similar reason the money cost of transportation of goods 
ought not to be counted, because it is a factor in the price of 
goods, and so is already counted in them. But the money cost 
of transportation of persons, or of travel, ought to be counted. 
Bonds are not to be counted because their prices depend upon 
the general rate of interest on the one hand and on the other 
upon the particular credit of each company, both of which fac- 
tors have nothing to do with the make-up of the exchange- 
value of money, although they are both affected by variations 
in the exchange-value of money, which therefore needs to be 
measured independently of them. 

The classes to be counted are, then, to be confined to so-called 
raw products and to those things which have been called fungi- 
ble, namely things sufficiently alike, every one in its own class, 
to replace one another with indifference on the part of their 
owners, which are things that can be meted out by weight or 
other measures, or by the piece. These things include not only 
raw products — grain, cattle, metals, etc. — but also many manu- 
factured products in a medium stage — steel, flour, yarn, cloth 
— and a few finished products — rope, some simple tools, bread. 



EXCLUSIONS AND INCLUSIONS 79 

and even ready-made clothing and shoes, which are now turned 
out in large quantities in almost uniform grades. All of these 
classes, which continuously provide a succession of very nearly 
similar individuals definitely measured and priced, are to be 
sought for and included. Perhaps two or three hundred such 
classes may be found, which number is not too large to be dealt 
with by a board of trained statisticians, collecting data and 
making calculations in the course of present time. 

§ 3. In trying to discover the ups and downs of the exchange- 
value of money in times past, it is impossible to find continuous 
price-lists ot so many classes, or information about the quanti- 
ties of the individual things composing them ; so that here we 
must confine our researches to a few. The few then must be 
chosen as samples. Now serviceable for samples are : (1) only 
the most important or staple products, (2) only those things 
whose prices are independent of each other, (3) only those whose 
prices are not subject to special causes of fluctuation ; and, fur- 
thermore, (4) some of these articles should not appear twice in 
the lists, once as a raw material, and again in the manufactured 
article made of it, while others appear only once. Some econo- 
mists have even advocated this " exclusive " method for present 
times. There is no good reason for doing so. This method is 
only a makeshift, the best we can do in reviewing past times. 
We can do better for present times, and should, therefore, seek 
to do so.^ Some of the early writers were content to measure 
the variations in the exchange-value of money by comparing it 
only with one other article, generally wheat, or the most com- 
monly used food product.- The line of progress has been to 
widen the range of the comparison more and more, until it 
reaches the utmost limit practicable. Now, it should be noticed, 

1 Slightly different is the position of those who advocate a " multiple standard " 
for contracts, or for regulating the exchange-value of paper money, composed of 
twenty, of a hundred, or of any other small number of classes, selected for their 
importance or from other motives. Their position is, not that this procedure 
yields the best measure of the exchange- value of money, but that it forms a much 
better standard than the standard resting on one or two metals. Still, while they 
are about it, they might as well have the largest " multiple standard " possible. 

2 So Locke, Adam Smith, Condillac, Say, D. Stewart, Storch, Cibrario, and 
others. 



80 ARRANGEMENT OF PARTICULAR EXCHANGE- VALUES 

the cauous suitable for the " exclusive " method have little or 
no application for the broad "inclusive" method.^ 

§ 4. In all cases only wholesale prices are to be employed. 
This is because they are the only ones practicable. Objection 
is often made that it is retail prices which give the exchange- 
value of money to the consumers. This is not altogether true, 
as the economic " consumer " is often the purchaser in whole- 
sale quantities, as the manufacturer, who consumes raw mate- 
rials ; and also many large institutions, hotels, governments, 
etc., mostly get their stores at wholesale prices. Although on 
some occasions retail prices do not follow the variations of 
wholesale prices, and may even vary slightly while those are 
stable, it is probable that the true average of retail prices fol- 
lows the ascertained average of wholesale prices much more 
closely than the attempt to reckon the average of retail prices 
would succeed in following the true average.^ 



II. 

§ 1. We have seen that when a class of commodities varies 
in price, all the individuals in it do so. Now if one class is 
larger than another, when they both vary in price, the variation 
of the larger class represents a variation of more individuals, 
and the variation of money in exchange-value in that class is a 
variation of it in more individuals, than is its variation in the 
other class. The particular exchange-value of money in a larger 
class is not necessarily a higher or greater exchange-value, but 
it is, so to speak, a wider or larger exchange-value, than its 
particular exchange-value in a smaller class. Therefore the 
variation of money in the larger class should count for more, 
or, which is the same thing, the variation in price of the larger 

^ The distinction is not observed in the following statements: "No article 
should be scheduled twice in different stages of manufacture," J. B. Martin, Gold 
versus goods, in the Journal of the British Association for the Advancement of 
Science, for the year 1883, p. 626 ; " We might count the wool instead of the things 
made of it (for of course we ought not to count both)," Marshall, B. 93, p. 373. 

* At all events, if retail prices are used, they must be used in a measurement 
by themselves. Wholesale and retail prices must never be mixed in one table. 
So Beaujon, B. 95, pp. 110-111, 114, 116. 



NEED OF WEKJHTING 81 

class should count for more. The making allowance for the 
sizes of the classes, which consists in assigning to each class its 
proper importance or weight in the calculation of the general 
variation of prices has been called " weighting," and the size 
assigned to each class has been called its " weight." These 
not very Avell chosen terms have become consecrated by usage. ^ 
We should notice that the idea of weight attaching to all classes 
refers to their influence upon the result in our calculations. 
The relative weights of the classes in our calculations are not 
relative weights of the classes per se. They are dependent upon 
the relative sizes of the classes per se. 

§ 2. The prices of the classes of things are quoted in the 
market lists on mass-units of various kinds and magnitudes, 
differently in different places and times, as it has been found 
convenient for merchants to bargain for (e. (/., iron by the ton 
and copper by the pound). The quoted prices of some things 
are therefore very much higher than those of others, without 
reference either to the preciousness or to the importance of the 
classes. If these prices be taken simply as they occur and be 
combined in a certain way, the operation, as will be shown 
later, is virtually that of weighting the classes according to the 
accidental height of their quoted prices at the first period, in a 
comparison of two or more periods. Thus the weighting here 
is purely accidental and haphazard, without any principle or 
reason for assigning more importance to one class than to an- 
other, except the chance of mercantile customs, which have 
grown up without reference to this subject. Such a method of 
calculating general exchange-value and its variations was em- 
ployed — of course without knowledge of what they were doing — 

^ The tei-ms have long been in use in the reduction of observations, especially 
in astronomy. They appear to have been introduced into economics by Jevons, 
who in his important first worli on our subject wrote : " It must be confessed that 
the exact mode in which preponderance of rising or falling prices ought to be de- 
termined is involved in doubt. Ought we to take all commodities on an equal 
footing in the determination ? Ought we to give most weight to those which are 
least intrinsically variable in value? [Cf. Malthus, above in Chapt. I., Sect. I., 
Note 8.] Ought we to give additional weight to articles according to their im- 
portance, and the total quantities bought and sold? The question, when fully 
opened, seems to be one that no writer has attempted to decide — nor can I attempt 
to decide it." B. 22, p. 21. 



B2 ARRANGEMENT OF PARTICULAR EXCHANGE- VALUES 

by one of the earliest investigators in this subject in the eight- 
eenth century, the French firxancier, Dutot, and in the century 
just elapsed even by two prominent economists.^ It was avoided 
in the middle and toward the end of the prior century by two 
other investigators, the Italian writer on monetary matters, 
Carli, and the English physicist. Sir George Shuckburgh Evelyn, 
the latter of whom introduced the practice of reducing all prices 
at one period, taken as basis, to 100 (i. 6., 100 money-units), 
whatever be the quantities of goods which are sold at this price 
in each class, or whatever be the number of times this money^s 
worth of goods of each class are sold during the periods com- 
pared. Here, though it is usual to say there is no weighting, 
there really is weighting, since all the classes are treated as if 
they were of the same size — there is even iveigJiUng, which is 
only next worst after the earlier haphazard weighting. It is, 
in fact, impossible to avoid assigning one weight or another to 
the classes reviewed ; wherefore it is plain that the proper thing 
for us to do is to distribute the weighting with the greatest care 
possible. 

§ 3. Not infrequently it has been asserted that it is shown by 
experience not to be worth the trouble to assign proper weight- 
ing to classes, as the so-called " unweighted " calculations 
(really with even weighting) yield results very little divergent 
— so it is claimed — from those reached with proper uneven 
weighting. Whether it is worth the trouble to be careful, even 
though the divergence were small, would seem to depend some- 
what upon the question who is to take the trouble ; for what 
would excuse a voluntary investigator surveying past events 
for a merely historical purpose would not excuse a board of of- 
ficial statisticians employed by the State for the practical pur- 
pose of providing a guide for contracts or for the regulation 
of money in the present course of time. But, although the gen- 
eral trend upward or downward in a series of years may be 
somewhat similar in the two methods, the particular results are 
often very divergent, as shown by the very calculations upon 

2 See Appendix C, I. As an example of the curious combinations this method, 
or want of method, may lead to, it maybe noticed [that in one of his calculations 
Levasseur allowed 1750 times greater influence to tin than to cotton ! 



NEED OP^ WEIGHTING 83 

which the other opinion has been founded. Thus in the com- 
parison given by Mr. Palgrave of the Economist series of " un- 
weighted " index-numbers and the "weighted" index-numbers 
calculated upon the same prices, we find the following contrasts : 



Year 
1880 


Evenly weighted 

87 


Unevenly weighted 
89 


1881 


81 


93 


1882 


83 


87 


1883 


79 


88 3 



Here the calculated movements of general prices go in ex- 
actly opposite directions in every sequence of years. Between 
the first and the second years, for instance, the Economist figure 
falls 7 per cent., and the "corrected" figure rises 4| per cent., 
— a difference of 12 per cent. Divergences of this sort are to 
be seen in every case where in a series of periods the same 
prices have been treated in both ways for comparison. By the 
same argument, therefore, by which it has been attempted to 
show the needlessness of uneven weighting, the need of it is 
proved.^ 

To assign the " weights " with perfect precision would involve 
a great amount of labor — principally in discovering the relative 
sizes of the classes ; for the mere introduction of the ascertained 
weights in the calculations does not much increase the labor. 
But to assign uneven weighting with approximation to the rela- 
tive sizes, either over a long series of years or for every period 
separately, would not require much additional trouble ; and 
even a rough procedure of this sort would yield results far 
superior to those yielded by even weighting. It is especially 
absurd to refrain from using roughly reckoned uneven weight- 
ing on the ground that it is not accurate, and instead to use 
even weighting, which is much more inaccurate. 

3 B. 77, pp. 329-330. 

* We shall, however, later find that both the methods used ia the above cal- 
culations are very defective, and especially so in the matter of affording com- 
parison between any two years neither of which is the basic year, so that much of 
their divergences maybe due to error even in the "corrected" figures. What 
difference would, then, exist in practice between the true method and the simple 
method is still an unknown quantity. But we shall later also find that in the 
proper way of forming a series of index numbers a slight error in every calculation 
may, in a wrong method, accumulate before long into large error. Therefore 
every contrivance to secure accuracy is imperative. 



84 ARRANGEMENT OF PARTICULAR EXCHANGE- VALUES 

§ 4. Still, although few of the practical investigators have 
actually employed anything but even weighting, they have 
almost always recognized the theoretical need of allowing 
for the relative importance of the different classes ever since 
this need was first pointed out, near the commencement of the 
century just ended, by Arthur Young/ But the method of 
measuring the sizes of the classes has been the subject of diverse 
views, and even the reasons offered for the most commonly 
adopted method have not been quite satisfactory. 

Arthur Young advised simply that the classes should be 
weighted according to their importance. Some early critics of 
the plan of judging the value of money " by its relation to 
the mass of commodities " objected that the importance of 
articles is different to different persons, and therefore there 
could be no one standard of this sort.*" Joseph Lowe yielded 
so much to this criticism as to recommend for different ranks of 
society different weightings, and consequently so many different 
standards ; but again he wanted one standard to be formed with 
weighting according to the importance of the classes for the 
whole community, as indicated by the total " values " con- 
sumed.'' That the classes should be weighted according to their 
relative total money-values, has become the prevalent doctrine, 
being frequently re-invented as if by instinct,^ and generally 
with the same explanation. One of the best forms in which 

s Young did so in opposition to Evelyn's method, which he condemned as 
manifestly wrong in counting the articles as equally important, B. 6, pp. 68, 70. 
There have, however, been relapses. Thus Porter, in B. 11, totally ignored un- 
even weighting, although his attention had been called to it by Tooke, who there- 
upon declared his table to be misleading because it allowed equal influence in the 
result to unimportant as to important articles. Evidence before the Select Com- 
mittee on Banks of Issue, 1840, q. 3615. Then a period followed in which even 
weighting was employed with little notice of the need of anything else, until 
Drobisch demolished Laspeyres' method for this neglect, B. 30, p. 145, cf. B. 29, 
pp. 32-33. Laspeyres admitted the theoretical need of uneven weighting, B. 26, 
p. 304 ; and since then attention has been paid to it by almost all writers. 

^Ricardo, Works, p. 401 ; Malthus, op. cit., pp. 119-120. So also Von Jacob, 
the German translator of Lowe, according to S. D. Horton, Silver and gold, Cin- 
cinnati, 1876, 2d ed., 1895, p. 39. 

■^ B. 8, Appendix, pp. 93-99. 

8 Tooke, loc. cit. ; Giffen, B. 45, q. 709 ; Sidgwick, B. 56, p. 66 ; Palgrave, B. 
77, p. 344 ; Sauerbeck, B. 79, p. 595 ; Marshall, B. 93, p. 372 ; Wasserab, B. 105, 
pp. 87, 89, 94 if. ; Westergaard, B. 110, p. 220; Fonda, B. 127, p. 160. 



NEED OF WEIGHTING 85 

this explanation has been presented is that the weighting then 
follows the importance of the articles to the " average con- 
sumer." ^ The standard of exchange-value now becomes what 
has aptly been called a " standard of desiderata." ^" 

To this view the objection has sometimes been made that it 
is too " subjective." An " objective " criterion has been in- 
voked by saying that ''more weight should be assigned to those 
commodities which, being circulated in greater quantities, make 
greater demand on the currency." " Here the measurement of 
the exchange-value of money is viewed in a peculiar manner. 
For no one would dream of measuring the exchange-value 
of any commodity by weighting the other articles in which 
its particular exchange-values vary merely by the quantities 
of them actually exchanged for that commodity. In order 
that one thing have exchange-value in another it is not neces- 
sary that the two should be actually exchanged for each other ; 
and the consideration by which the size of the other thing's 
class is to be judged cannot possibly be confined to the quantity 
of it exchanged for this thing. There is nothing peculiar here 
in the case of money. We are not measuring the demand for 
money, but the general exchange-value of money. This doc- 
trine, therefore, is not so good as the one it seeks to supersede. 

In regard to the question of objectivity and subjectivity, it 
may be noticed that these terms are used in two senses. One 
of them is that " objective " refers to what is universal to all 

^Marshall, B. 93, p. 372. — The "average coiisumer,"of course, is the whole 
corumunity, and uot a few samples of" normal families." The method of ascer- 
taining the "average consumer" by means of family budgets (especially if con- 
fined to those of day laborers, as often done — and advised by Pomeroy, B. 135, p. 
332) is certainly not so accurate as the method seeking the total volume of goods 
in trade. The investigation of family budgets, introduced by Eden and elabo- 
rated by Engel, is very important for sociological studies, but hardly interests us 
here. The statistician who has used them most in our department is Falkner (see 
especially B. Ill, pp. XL-LV), who defends them in B. 112, pp. 63-64 and B. 113, 
pp. 269-270. The two methods are discussed by Taussig, B. 121, pp. 24-25. 

1" Horton, op. cit., p. 35 ; The silver pound, 1887, pp. 3-4. That such a standard 
is one of the objects of these enquiries has been asserted by the British Association 
Committee, First Report, B. 99, p. 249 n. 

11 Edgeworth, B. 66, p. 139. This would lead to a slight change in the weight- 
ing, as perceived by Foxwell, who, according to Edgeworth, B. 63, p. 135, wanted 
weighting to be assigned according to the values of tlie classes multiplied by the 
numbers of times the articles in tliem are sold before being consumed. 



86 ARRANGEMENT OF PARTICULAR EXCHANGE- VALUES 

men, in distinction from what is peculiar to individual men as 
" subjective." In this sense the above measurement of the 
sizes of the classes according to their importance for the whole 
community is objective. Subjective would be the many stand- 
ards, one for each person according to the importance of the 
classes to him individually. Still we desire a measurement, and 
a reason for it, which are "objective" in the other sense of 
being out of dependence upon persons. The case is somewhat 
like the measurement of heat. Heat is subjective, regarded as 
a feeling in men, though it be in all men. Its variations are 
also subjective as being differently felt by different persons. 
But we measure heat without regard to the susceptibilities of 
individual persons. And more still : we measure it without 
regard to its universal influence upon all men. A being who 
never had the sensation of heat, could measure heat. Could 
not a being who knows not the idea of importance rightly 
measure out the proper weighting to be assigned to the classes 
of exchangeable articles ? 

§ 5. A few economists have held that weighting should be 
according to the relative masses that are consumed of the differ- 
ent classes, all the masses being measured by the same common 
mass-unit, a unit of weight being generally preferred. ^^ This 
position claims to be the truly objective. It has the fault that 
it may be objective in two distinct ways, neither of which has 
any superiority over the other. For if we measured the masses 
of the classes by a common measure of capacity, the relative 
sizes of the classes would turn out differently. It may be re- 
plied that it is weight which gives the mass proper, or quantity 
of matter, contained in commodities. Tlie counter-reply is that 
capacity is the measure of volume, or size in space ; and there 
is no reason why the quantity of an inscrutable thing called 
matter should be more considered, in economics, than the quan- 
tity of visible space which the objects occupy. We compare 

12 So Drobisch, B. 29, pp. 30, 35, B. 30, pp. 148, 153 ; Lehr, B. 68, p. 11 ; Lind- 
say, B. 114, p. 26 ; and apparently the British Association Committee, First He- 
port, B. 99, pp. 249-250; and Edgeworth, B. 59, p. 264. But statements on this 
matter are often imprecise, the ambiguous term " quantities " being used. We 
shall also see that in some cases weighting which appears to be by masses is really 
by total money-values. 



WEKJHTIXG EXPLAINED 87 

some exchangeable objects by volume or bulk, and consider it 
an advantage that they be heavy. We compare other ex- 
changeable objects by weight, and consider it an advantage that 
they be large (and therefore light). We never compare all 
things either by weight alone or by bulk alone. And further- 
more, some exchangeable objects, such as land, have neither 
weight nor volume, while other things, such as railroad tickets, 
no one would think of sizing according to their weight or vol- 
ume ; and some gases, which now are commercial articles, and 
sold by capacity at a certain pressure, it would be difficult to 
bring into line with solid or liquid things, since the pressure 
chosen is arbitrary. But to this position a more direct objec- 
tion will be noticed presently. 

III. 

§ 1. Such is a brief history of the question of weighting 
brought down to the present day. It is obvious that the ques- 
tion has not been thoroughly discussed. Even the nature of 
weighting in general has rarely been understood, the term it- 
self being misleading ; and the special difficulties concerning 
weighting in our subject have never been pointed out, where- 
fore they have never been overcome. 

We shall see that another question which naturally arises in 
our subject is a question of averaging the variations of exchange- 
values. Now weighting is a question connected with averag- 
ing, and though even weighting is the more common in simple 
theoretical problems, in practical problems occasion for uneven 
weighting is' always likely to present itself. The question of 
weighting must be treated first, because weighting of some sort 
is a prerequisite in all averaging. 

The nature of weighting may be illustrated by a simple ex- 
ample. We may suppose that two proud fathers, each with 
three sons, dispute as to which has the taller sons, and proceed 
to measure and to average them. The one finds a different 
figure for the tallness of each son. He simply adds up the 
three figures, and divides by three. He has used even weight- 



88 ARRAXGEMEXT OF PARTICULAR EXCHAXGE-VAEUES 

ing, allowing equal importance in the calculation to each meas- 
urement. The other, let us suppose, finds two of his sons to 
be equally tall and the other differently tall. He then has only 
two different figures. The one represents the tallness of two 
sons, the other the tallness of one son. If he adds these two 
figures and divides by two, the result would be wrong. If he 
notes down the figures for every son separately, although the 
same figure occurs twice, and adds these three figures and di- 
vides by three, he would get the right result, and be doing ex- 
actly what the other did — he would be using even weighting, 
with three figures. But if he employs only the two distinct 
figures, but takes the one representing the two measurements 
twice, by multiplying it by two, and adds this doubled figure 
to the other single figure, and divides by three, he would be 
using uneven weighting ; for he would be allowing twice as 
much influence upon the result to the one figure as to the other. 
The result is the same as 'if he employed even weighting with 
addition of three figures. In this simple example the differ- 
ence in the labor of performing either of these operations is 
not great. But in more complex matters it may amount to a 
great deal. The operation of first multiplying all the similar 
figures by the number of times they occur and adding their 
products to one another and to the figures that occur only once, 
and dividing by the total number of repeated and single figures, 
is a simpler one than that of adding all the individual figures. 
Thus uneven weighting, though appearing to be more difficult 
than even weighting, is really a means of simplifying and 
abridging the calculation. 

Weighting, then, consists in allowing for the number of in- 
dividuals which possess a common attribute in certain quantities, 
which are being averaged. Even weighting is employed when 
every individual is measured, and its measurement, whether the 
same or different from others, is separately employed in the cal- 
culation. Uneven weighting is employed when equal measure- 
ments are treated as one, for convenience merely, and yet allow- 
ance is made for the number of times they are repeated. Or 
uneven weighting is to be employed, again, when we make only 



AVEIGHTIXG EXPLAINED 89 

a few observations on different classes of indiviclnals, and know 
(or assnme) that the other individuals are like those measured ; 
for then the quantity of the attribute possessed by each indi- 
vidual in the class is really to be repeated the number of times 
it is possessed by an individual, that is, by the number of in- 
dividuals in the class. Only if the numbers of individuals in 
all the classes happen to be the same, can even weighting be 
properly employed. 

§ 2. Now in most subjects of averaging, the nature of the 
individuals in each class is directly given, wherefore the finding 
of their numbers is not difficult — at least theoretically speaking. 
Thus, in the above example, the three sons in each set were the 
individuals in question. Or in the wider case of measuring the 
average tallness of the population of a country, after getting 
the average tallness for different districts, we should weight the 
calculation of the general average of these by the numbers of 
persons in each district. This is because, by the nature of the 
problem, every full-grown person, be he prince or pauper, is an 
individual equally important with every other as a factor in 
determining the result we are seeking. 

In the subject of averaging exchange-values the difficulty 
which confronts us is apparent. The individuals with which 
we are dealing in every class of commodities whose prices are 
reported, are not directly given ; for, as already remarked, it is 
purely accidental what the actual individuals are which have 
prices recorded against them. The price of wheat is now in 
America generally reported in bushels. In England it used to 
be reported in quarters. If these two systems be applied to 
the same mass of wheat, the former would report the price of 
eight times as many individual portions of wheat as the latter, 
the portions referred to in the latter being eight times as large 
as those in the former. It is obvious that the number of times 
the price of a bushel of wheat is repeated has no more right to 
be the 'Sveight " of the wheat-price in our calculations than 
has the number of times the price of a quarter of wheat is re- 
peated, or reversely, or than the number of any other accident- 
ally chosen mass-unit of wheat. And so with the prices, and 



90 AERAXGEMENT OF PARTICULAR EXCHAN&E- VALUES 

the numbers of individuals which bear them, in all other fungi- 
ble articles. It is, therefore, a problem what are the individuals 
we are dealing with. This question must be answered before 
we can find the sizes of the classes, which sizes depend upon 
the numbers of individuals there are in each class. 

§ 3. Here a certain position above reviewed seems to offer aid. 
This is the position that we ought always to take the prices of 
the same mass-unit, preferably one of weight, applied to all com- 
modities, and to judge the sizes of the classes by the numbers 
of this mass-unit they contain. According to this doctrine, the 
individual in all classes is the same mass-quantity, or the same 
space-quantity, and the relative sizes of the classes are according 
to their relative total weights or volumes. The former idea 
being adopted, the " weights " of the classes in our calculations 
are to be literally according to the weights of the classes. This 
is, evidently, a very convenient doctrine. 

Against it some objections have already been urged. But the 
principal one has not yet been noticed. This is that weighting 
so determined would be according to a different attribute in the 
things from the one we are measuring. It would be like weight- 
ing the calculation of the average tallness of the three sons, 
above cited, by the weight or girth of each of the boys. It is 
plain that the relative weight or volume of two classes of com- 
modities does not indicate the relative importance of these t^vo 
classes. We have, however, agreed not to judge the sizes of 
classes by their importance. But it is equally plain that the 
relative weight, or volume, of two classes does not indicate their 
relative sizes from the point of view in which we have the 
classes under consideration. Their relative weights indicate 
their sizes judged by weight. Their relative volumes indicate 
their sizes judged by volume. What indicates their sizes judged 
as things possessing exchange-value ? Evidently only the rela- 
tive amounts of the exchange- values of their whole quantities.^ 

1 Of course in the above examples we would not weight the figures according 
to the tallness of the boys, or according to the total tallness of the people in the 
different districts. This is because in those examples the individuals are given, 
being discrete wholes. The proper comparison is with averaging in the case of 
things with uniform substance that are found in different sizes. In averaging, 



WEIGHTING EXPLAINED 91 

To prove this, we must review the nature of the subject for 
whicli we are seeking the weighting. 

§ 4. What we are seeking ultimately is the combined result 
on the general exchange-value of something when we know the 
variations in its particular exchange-values. We have seen 
that when a particular exchange-value of the thing rises to a 
certain height, this rise may be compensated by another of its par- 
ticular exchange-values falling to a certain extent, although we 
have not yet found the pro])ortion between these balancing vari- 
ations. But we have also seen that if two of its other ])articular 
exchange-values fall instead of only one, these would each have 
to fall to a less extent to counterbalance the one rise (in Propo- 
sition XXX.). We w^ant, then, to know what constitutes one 
particular exchange-value, and what constitutes two particular 
exchange-values — that is, two particular exchange-values that 
are twice as large (or wide, so to speak) as the one. Thus far 
Ave have merely treated of a particular exchange-value as one if 
it be merely an exchange- value viewed in relation to one other 
exchange-value, without regard to the size of either of the two 
classes. This we now find to be not quite adequate. We find 
the need of another distinction. Let us keep the term " ])artic- 
ular-value " in its previous usage, and introduce for what we 
now want the term " individual exchange-value." A j)articular 
exchange-value of a thing is its exchange-value in one other class 
of things ; an indiA'idual exchange-value is its exchange-value 
in one other thing. An individual exchange-value relates to a 
particular exchange-value very much as a particular exchange- 
value relates to general exchange-value ; for as many particular 
exchange-values compose one general (or generic) exchange- 
value, so many individual exchange-values comjjose one partic- 
ular (or specific) exchange- value. Our question then is, What 
constitutes one individual exchange-value, whose variation is to 
be counterbalanced by an o^jposite variation in some other in- 
dividual exchange-value, and by a lesser variation in other two 
individual exchange- values ? 

for instance, the yields of different fields (in which the superficies is the element 
of importance) in order to tind the average yield of a country, we should count 
every field, not as one individual, but according to the relative extent of its 
superficies. 



92 AEEAXGEMENT OF PARTICULAR EXCHANGE- VALUES 

Now when one thing has a certain exchange- vakie in another 
thing, it is equal in exchange-vakie to the quantity of that other 
thing by which its exchange-vakie in it is measured. Hence if 
the thing lA is equivalent to 6B and to cC (these capital letters 
referring to mass-units, and the small letters to numbers), we see 
that when we speak of A's rise in exchange-value in B being 
compensated by its fall in exchange- value in C, we have in mind 
its rise froni equivalence to 6 B and its fall from equivalence to 
c C, wherefore these are the two individual exchange- values we 
are comparing with each other at that period. Therefore also the 
two individual things in the classes B and C with which we are 
comparing lA are 6B and cC — its equivalents at the period in 
question. Similarly if 1 A be equivalent to c D, the economic in- 
dividual in the class D with which Ave are comparing lA is cD. 
Suppose the rise of 1 A to equivalence with b'b B is compensated 

by the fall of lA to equivalence with -, cC {b' and c' being 

certain figures larger than unity). Then if A falls in D also, 
the balance is disturbed. Therefore, for the balance to be main- 
tained, when A falls in C and in D equally, it must fall in 

each less than by -, . Now suppose the individuals c C and c D 

are exactly alike, so that we put them in the same class and 
refer to them by the same term, say Z. Then when A falls in 
Z, it falls both in C and in D equally. Therefore, to counter- 
balance its rise in B, it must fall less in Z than it would have 
to fall in C alone or in D alone.^ It is obvious that the class 
Z is larger than, and just twice as large as, the class B, at the 
period in question, because it embraces twice as many individuals 
as the class B, since it embraces two, namely c C and c D, while B 
embraces only one, 6 B. And so if B and Z are variously large 

2 Proposition XXX., it should be noticed, may apply both to other classes 
and to other individuals. It may mean that when the compensatory changes are 
in a greater number of (equal) classes, they must be smaller, and when they are 
in a greater number of (equal) individuals (in the same class), they must be 
smaller. The reason is that in the former case as in the latter the compensation 
is distributed over more individual things. It is plainly indifferent whether the 
individuals, their number being given, be all in one class or in many classes. 
The important factor is, not the number of classes, but the number of individuals. 



WEIGHTING EXPLAINED 93 

classes, their relative sizes will be determined by the numbers of 
6 B's and of c Z's they contain — that is, the numbers of equivalents 
to lA. Generalizing still more, we see that it is the number of 
equivalents to lA in any class, which constitutes its relative 
size, these equivalents to lA being the economic individuals in 
them corres])onding to the posited economic individual lA. 
And, of course, whatever be the quantity lA, a large or a small 
mass-unit, the proportions of the numbers of equivalents to lA 
in the other classes, and consequently of their sizes, will be the 
same relatively to one another and to the class A. Hence if 
instead of lA we use IM — a money -unit, say (me dollar, — then 
the relative sizes of all the classes of commodities are propor- 
tioned to the " dollar's worths " they C(^ntain, which are indicated 
by the total money-values of these classes. 

All this reasoning is obviously independent of any considera- 
tion of the mass-quantities of 6 B and of c C, supposed to be 
equivalent to lA, or of the mass-units in wdiich they are ex- 
pressed. If lA is one pound of wheat, and this happened to 
be equivalent to one pound of butter as 6 B and to one pound of 
wool as cC, this would in no wise improve the reasoning, which 
M^ould be the same if the one pound of wheat happened to be 
equivalent to two pounds of butter and to seven and three- 
quarter pounds of wool. On the other hand, if this last hap- 
pens to be the case, it is plain that one pound of butter and one 
pound of wool have no claim to be considered economic indi- 
viduals ('(jmpared Avith one pound of wdieat. As masses, one 
pound of butter and one pound of wool are individuals compared 
with one pound of wheat, because equal to it in weight. But 
we are not dealing with masses. We are dealing with exchange- 
values. And the exchange- value of one pound of wheat being 
taken as our given individual, it is only the quantities of butter 
and of wool whose exchange values are equal to this exchange- 
value, at the period in question, that are individuals compared 
with it, in our economic point of view. 

Or if we compare classes with money, the state of things is 
still plainer. Suppose copper is worth ten cents a pound and 
iron one cent a pound. It is evident that when we compare a 



94 AERANGEMENT OF PAETICULAR EXCHANGE- VALUES 

variation of the exchange-value of one dollar in copper with a 
variation of the exchange-value of one dollar in iron, we are 
comparing its variation relatively to ten pounds of copper and 
to a hundred pounds of iron. And if during a certain period 
the masses of copjDer and of iron with which we are dealing are 
ten million pounds of the former and fifty million pounds of the 
latter, the relative sizes of these classes — the relative numbers 
of individual exchange- values of IM m each of these classes, 
that vary when its price-quotation varies — is, at the period in 
question, not as ten for copper to fifty for iron (or 1 to 5), but 
as one hundred for copper to fifty for iron (or 2 to 1). 

It will be noticed that this explanation does away with de- 
pendence upon the idea of importance. The sizes of the classes 
are not measured by their importance to the consumers. They 
are measured by the numbers of equivalents they contain, these 
equivalents being the economic individuals, and the sizes of the 
classes naturally being according to the numbers of individuals 
they contain. It is true that the importance of the classes is 
measured in exactly the same way, so that the relative sizes of 
the classes go hand in hand with their relative importance — and 
we may continue to speak of their sizes being according to their 
importance. But it is not directly by their importance that the 
measurement of their sizes is made. It is made by the measure- 
ment of the above equivalent individuals they contain. This 
measurement is just as objective as the measurement by the 
number of equiponderant individuals they contain. 

§ 5. It should be noticed, further, that the size of the class 
whose exchange-value is being measured is of no consequence, 
and therefore need not be considered, except only in the one 
case when we are seeking to measure a variation of its exchange- 
value in all things. For if we are seeking to measure its vari- 
ation in exchange-value in all other things, or if we find that it 
has remained constant in exchange-value in all other things 
(and consequently in all things), we have to consider only the 
relative sizes of the other classes, the size of the class in ques- 
tion having no influence in the result. Thus in measuring the 
variations in the exchange-value of money, we are dispensed 



SOME DETAILS IX WEIGHTING 95 

from enquiring about the size of the class money itself, so long 
as we confine ourselves to measuring its variations in general 
exchange-value in all other things. To this we shall confine 
ourselves for the greater part of our researches, and shall treat 
of the more difficult problem of measuring variations in the 
general exchange-value of money in all things only at the end 
of our work, where, too, we shall consider whether such a 
measurement is needed or not. 

lY. 

§ 1. The number of individuals or equivalents in each class 
during a given period is, of course, not to be measured by the 
number of them in existence at any one moment. For articles 
are variously durable, and the stocks on hand at any moment do 
not represent the relative quantities in which they are ordinarily 
used. These quantities can only be ascertained by taking all 
the quantities brought into trade during a period sufficiently long 
to cover all the ups and downs of the stocks — at least a year as 
being the shortest natural cycle. If land be one of the things 
taken into account, not its total money-value, nor even the total 
money-value sold during a year, but its annual rental is the item 
to be compared with the total money-values of the commodity 
classes.^ 

§ 2. It is a question whether the total money-values of com- 
modity classes to be considered are those of the quantities an- 
nually produced or those of the quantities annually consumed ; 
for in every coimtry importation and exportation disturb the 
equality between production and consumption in the case of 
many articles. Perhaps it would be best to count both — that is, 
in any one case either the total product or the total consumption 
according as the one or the other is the greater. Thus in the 
United States we should count all the wheat produced and all 
the tin consumed, and in England all the wheat consumed and 
all the tin produced. There is overlapping here ; but this is 
right, because the English and the American economic worlds 

1 For a similar reason it is only the variations of rent, and not the variations 
of the price of land, that are to be counted. 



96 ARRANGEMENT OF PARTICULAR EXCHANGE-VALUES 

do overlap. This double counting, which takes place in the two 
measurements, could be avoided by making the measurement of 
the exchange-value of money in the two countries together, if 
that be possible ; but then there would be similar overlapping 
with other countries, which could be avoided again only by mak- 
ing the measurement for the whole world at large — for then we 
should have to count only the total quantities produced or con- 
sumed, without counting exports and imports twice.^ Of course, 
the total money-values must not be confined to the total money- 
values sold. A bushel of wheat consumed by a farmer is as 
valuable as a bushel of wheat which he sells. 

§ 3. Articles that pass through various stages and appear in 
each under a special name — as hemp, rope ; wheat, flour, bread ; 
wool, yarn, cloth, clothing, — should not be counted as a distinct 
quantity in each stage. If that were done, as some articles ap- 
pear in more forms than others, the sizes of these would be 
unduly magnified. One method of treating these things would 
be to group together all the various stages, and to assign to this 
group a size according to the total value of its highest forms (or 
the highest counted). Another would be to confine attention to 
two stages only. But in the second stage there might be several 
branches, according to the kinds of things the raw material is 
made into. 

§ 4. To find with accuracy the total money-value and the 
average price of any commodity class during a given period is a 
statistical task of considerable difficulty ; but the theory of the 
operation is simple. Of all the prices reported of the same kind 
of article the average to be drawn is the arithmetic ; ^ and the 
prices should be weighted according to the relative mass-quan- 
tities that were sold at them. For example, if there is sold 
twice as much cloth in September as in July, the price in Sep- 

2 Some statisticians have counted only these overlappings, and have weighted 
the classes according to their relative sizes in the country's exports and imports. 
If this weighting be employed merely in an attempt to measure the volume of 
foreign trade, as done by Giffen and De Foville, less objection can be found, 
.But if it be employed in a measurement of the general exchange- value of money, 
it is a very defective kind of weighting. Its defect has been noticed by Nasse, B.. 
104, p. 332. 

3 Jevons, B. 15, pp. 41, 43 ; Walras, B. 61, p. 6. 



THE QUESTION OP PERIODS 97 

tember sliould be given twiee as mueli weight as the price iu 
July/ 

V. 

§ 1 . As yet we have examined the sizes of the classes only at 
a single period. But in any enquiry into the variation of the 
general exchange-value of anything we are concerned with at 
least two periods. Between the two periods not only the par- 
ticular exchange-values may have varied, but also the sizes of 
the classes. And the sizes of the classes vary partly because of 
the very variations in the particular exchange- values whose com- 
mon variation we are attempting to measure. The difficulty iu 
the subject of weighting, in the case of averaging exchange- 
values, is only commencing. 

The older writers, from Arthur Young on, who touched upon 
weighting, recommended merely a rough weighting according to 
the importance of the classes as vaguely measured, by their 
relative total money-values, or by masses, over a number of 
years, although a few, using custom-house returns, actually did 
use varying weighting, without understanding its nature.^ The 
need of paying attention more minutely to the weighting of each 
period compared was first pointed out about thirty years ago by 
the German philosopher and mathematician, Drobisch. Drobisch 
was the originator of the idea that we should weight the classes 
according to their actual physical weights, and he confined his 

* E. Segnitz gave this formula for finding one average price, 

3 + g'+3"+--- ' 
in which p^'p" represent the prices at which respectively qq'q" quantities of the 
article sold during the period in question, Ueber die Berechnung der sogenannten 
Mittel sowie deren Anwendung in der Statistik und anderen Erfahrungswissen- 
schaften, Jahrbiicher fiir Nationaloekononiie und Statistik, 1870, Band XIV., p. 
184. E. Heitz, though not approving of it himself, says this method is employed 
in the statistical bureaus of several German states (Wiirtemberg, Bavaria, Gotha, 
and partly in Prussia and Hannover), Ueber die Blethoden bei Erhebung der 
Frcisen, in the same Jahrbiicher, 187(5, Band XXVII., pp. 347-351. This use of 
uneven weighting has been recommended also by Marshall, B. 93, p. 374, and the 
British Association Committee, First Report, B. 99, pp. 250-251. — Weighting ac- 
cording to the length of time each price lasts was recommended by Beaujon, B. 
95, p. 116. 

iSee Appendix CIV §2 (2). 

7 



98 ARRANGEMENT OF PARTICULAR EXCHANGE- VALUES 

attention to diiferences at the two periods between, the mass- 
quantities of the classes. Considering the single case of a com- 
parison between two contiguous periods, he raised the following 
questions : — Shall we employ in our calculations the mass- 
quantities of both the periods separately ? or only the mass- 
quantities of one of the periods ? and in the latter case shall it 
be those of the first period, or those of the second ? or, again, 
shall we combine these two systems, and employ a single one 
that is a mean between them ? Between these questions Dro- 
bisch decided in favor of the first course suggested. In doing 
so he originated what may be called a method of double weight- 
ing. This consists in drawing an average of the prices at each 
period separately, and at each period on the mass-quantities of 
that period ; and in then comparing these averages. At each 
period compared the mass-quantities may be measured in dif- 
ferent ways. The way Drobisch chose was, as we have seen, to 
measure them by the number in every class of a common mass- 
unit. Upon the method of double weighting he decided on the 
ground that the mass-quantities of neither of the two periods is 
preferable the one to the other,^ and that there is no better reason 
for the mean between the two,^ apparently considering that the 
course he adopted was the only one not exposed to objections. 
Drobisch has found but few followers in the use of double weight- 
ing, though some involuntary ones are among them.* His posi- 
tion was immediately attacked by Professor Laspeyres, for a 
reason which will be noticed later — a reason, however, which is 
valid, not against his use of double weighting, but against his 
use of a common weight-unit ; and a little later by Paasche, on 
the ground that we must use the mass-quantities of only one 
period so as to get the variation in the sums of money needed to 
purchase the same mass-quantities at both the periods.^ And a 
similar reason has been given more recently by Professor Falk- 
ner, namely that " we must compare like with like," ^ — although, 

2 B. 29, p. 39. 

3 B. 31, pp. 423-425. 
■* See Appendix C V. 

5 B. 33, pp. 171-173. 

6 B. 112, p. 63. 



THE QUESTION OF PERIODS 99 

if the like does not exist at both the periods, the comparison 
may appear somewhat forced. Laspeyres recommended using tlic 
mass-quantities of the first period, Paasche those of the second." 
The mean between these two positions may be either by making 
two calculations, the one on the mass-quantities of the first 
period, the other on those of the second, and then taking the 
(arithmetic) mean between the two results ; or by making a 
single calculation with mass-quantities that are the (arithmetic) 
mean between those of the two periods. The last has sometimes 
been recommended, though only half-heartedly.'^ Thus every 
obvious position seems to have been occupied. Amidst all this 
diversity of opinion Professor Sidgwick has asserted that as we 
cannot decide in favor of any of these methods, the use of the 
mass-quantities of the one period being as good as of the other, 
and the mean between them having no " practical significance," 
and as the answers obtained by each method may differ, we can 
therefore make no one authoritative measurement of the varia- 
tion of money's general exchange-value.'' And the same criti- 
cism has recently been reaffirmed by Dr. Wicksell, who says 
that the problem is insolvable unless it happens that the same 
result is yielded by using the mass-quantities of each period 
separately, the use of the mean bet^veen the mass-quantities at 
the two periods having nothing but a " conventional meaning." ^^ 
The truth is, however, that these are not the only positions 
that present themselves, and the subject has hitherto been treated 
most inadequately. Like Drobisch, the writers who have no- 
ticed this question have considered only the diffe'rence at the two 
periods between the mass-quantities. But it is strange that per- 
sons who have not followed Drobisch in adopting mass-quanti- 
ties as the test of weighting, and who have even asserted, many 
of them, that weighting should be according to the total exchange- 
values of the classes, when they come to discuss the question 
concerning divergent weights at the two periods, should confine 

"^ Pasche has been followed by v. d. Borght aud Conrad, and without depend- 
ence by Mulhall, by Sauerbeck at times, and, following the latter, by Powers. 
See Appendix CIV §2 (2). 

8 See Appendix CIV § 2 (3). 

3 B. 56, pp. 67-68. 

" B. 139, pp. 8-9. 

L.ofC. 



100 ARRANGEMENT OF PARTICULAR EXCHANGE-VALUES 

their attention to diiferences that may exist between the mass- 
quantities at the two periods/^ Changes in the mass-quantities, 
and changes in the weights, of the classes are two distinct things, 
and the former are of importance only as they affect the latter. 
Since no one has so much as noticed the question in jts proper 
aspect, we are left to our own devices, and must investigate the 
subject ah ovo. 

§ 2. There are three variables : (1) the exchange-values, or 
prices, of given mass-units of many classes of commodities ; (2) 
the total mass-quantities of these classes ; (3) their total ex- 
change-values, or money-values. The first are the ones we are 
trying to average. The second and third are factors in the 
weighting of the classes in our averaging of their exchange- values 
or prices, or of the variations of these. Yet the third itself is 
dependent upon the first two, being their product. As we are 
now examining only the factors that enter into the weighting of 
the classes, we are not concerned with the first variations except 
so far as they affect the third. 

With the two variables that enter into weighting we have, 
then, two distinct elemental cases, and a third composed of these 
two. The first, and simplest, is when the total sums spent on 
every class remain constant over both the periods, so that the 
relative sizes of the classes remain the same at both the periods, 
the variations of the prices being counteracted by inverse vari- 
ations of the mass-quantities. The second is when the mass- 
quantities in every class remaiu constant over both the periods, 
so that the sizes of the classes vary from period to period directly 
as the prices. The third is when both the mass-quantities and 
the total sums spent on them vary from period to period. 

§ 3. I. In this simple case the relative importance, or the 
relative sizes, of the classes remain constant at each of the 
periods, whether the exchange-value of money be the same at 
both the periods or not. It would therefore seem natural to 
treat the classes as having, in our averaging of their price- vari- 
ations, the relative sizes that they have at each of the periods. 

11 Giifen and Palgrave are exceptions. The former used weighting according 
to the money- values of a single period, the latter according to those of every later 
period. But each has done so without discussing the question. 



THE QUESTION OF PERIODS 101 

Thus, for example, if at the first period the total money-value 
of the class A is twice as great as the total money- value of the 
class B, and if at the second period each of these classes has the 
same total money-value as before, — or even if they have both 
varied in their total money-values, but in the same i)roportion, 
so that the class A still has twice the money-value (and conse- 
quently t'wice the exchange-value, as may be deduced from 
Proposition VII.) as the class B, — we should look upon the class 
A as twice as large, or as containing twice as many economic 
individuals, as the class B. 

Yet each of the economic individuals in these classes would 
consist at each period of a diiferent mass-quantity of the article. 
When this is perceived, we may hesitate to adopt this position. 
We may best discuss this question, however, when we are ex- 
amining the more general and more complex cases where both 
the total exchange-values and the total mass-quantities are vari- 
able. 

§ 4. II. The next simplest case is when it ha[)pens that the 
mass-quantities of the various classes remain constant over both 
the periixls, and their prices varying, their total money-values 
are different at each period. It is plain that this suppositional 
case escapes all the questions that have hitherto been raised on 
this subject, above reviewed ; and yet the question of weighting 
is still before us. For the sizes of the classes are different at 
each of the periods directly according to the variations in their 
relative prices. Hence we have alternative positions like those 
already noticed, except that here there is no opportunity for 
double weighting. Shall we use the weighting of only one of 
the periods ? and then shall it be that of the first or that of the 
second period ? or shall we use a mean between the two sys- 
tems? And there is still another possible position. Shall we 
use only the smaller total money-value that occurs at either 
period as the weight of the class for both the periods together, 
eliminating what is in excess at one of the periods ? 

Plainly there is no reason why in general we should choose 
the weighting of one of the periods in preference to that of the 
other. And there is good reason for rejecting each of them — 



102 ARRANGEMENT OF PARTICULAR EXCHANGE-VALUES 

in general, unless later discussion of the averages shows that the 
one fits one of the averages and the other another. For if one 
class happens to be more important, or larger, than another at 
the first period, and at the second, through a fall in the price of 
the former and a rise in the price of the latter, is less impor- 
tant, or smaller, it would be absurd to take either of these con- 
ditions as alone representative of the proper relationship be- 
tween these classes. Or if two classes are equally large in total 
exchange-value at the first period and the one becomes larger 
and the other smaller at the second period, and if two other 
classes, at the first period the one larger and the other smaller, 
become equal in size at the second period, it is plain that we 
have no better right to treat the classes in the one of these pairs 
as equal in size than to treat the classes in the other as such ; 
but it is impossible to do so with both, if we are to use the 
weighting of only one period — all which shows that the classes 
in neither of these pairs are to be treated as equal, yet not so 
unequal as in each they are at one of the periods.^^ 

§ 5. Therefore we must use either a mean between the weight- 
ing of the two periods, or the weights partly of the one and 
partly of the other, picking out the smaller weight of a class at 
either period. The reason for entertaining such a position as 
the last is obtained from the preceding Chapter. We have seen 
a desideratum to be that we should have the same or a similar 
whole at both periods, in which to compare the relationship be- 
tween a part and the whole of which it is a part. An economic 
individual being an exchange- value quantum rather than a mass 
quantum, it may be maiatained that we ought to eluninate all 
those individuals which exist at one period alone, which can be 
done by taking the total exchange-value that is the smaller at 
either period. The mass-quantities being supposed in all cases 
constant over both periods, if the class A rises in exchange- 

^2 It may be objected that it is only the individuals at the first period that 
vary, the individuals at the second period being merely a result of the variation of 
the former. But really the variation is from the individuals at the fii'st period to 
the individuals at the second period ; and the number of individuals that have 
varied is not the number existing at the first period any more than it is the 
number existing at the second. 



THE QUESTION OF PERIODS 103 

value between the first and the second period, its total exchange- 
value increases, and only that M^hich it had at the first period 
should be considered ; and if the class B falls in exchange- 
value, it contained at the first period more economic individuals 
than at the second, and so only those existing at the second 
period should be considered, 

A practical objection to this position is that the economic indi- 
viduals at each period should be the same, that is, have the same 
exchange-value, or be equivalents to a money-unit having the 
same exchange-value at both the periods ; but whether the 
money-unit has the same exchange-value at the second period 
as at the first, is the question to be determined. Merely to 
compare the total money-value of the classes at the two periods 
is not to compare their total exchange-values ; yet that is all we 
can do till we have found the constancy or variation of the ex- 
change-value of money. It is plain that if the exchange-value 
of money be falling, i. e., if prices in general be rismg, the 
smaller total money-values might all, or most of them, occur at 
the first period, and then in selecting them for our weights, we 
should be taking merely, or mostly, the weights of the first 
period ; or if the exchange-value of money be rising, i, e., if 
prices in general be falling, the weights might be mostly tho^e 
of the second period. Or between one country and another, 
where, the mass-quantities being the same, the levels of prices 
are considerably different, we might be really taking the weights 
almost altogether from those of one of the countries alone, namely 
the one in which the level of prices is the lower. It is perhaps 
possible that we might approximate to the ultimate result by 
several stages of approach. Assuming at first that money is 
stable, we should take the total money-value as representing the 
total exchange-values of the classes, and, choosing the smaller 
ones at either period, should use these as the weights of the 
classes ; then if the result showed constancy in the exchange- 
value of money, we should rest content ; but if it showed a vari- 
ation, we should take this as a means of correcting the estimate 
of the total exchange-values of the classes at the second period 
compared with ^vbat they were at the first, and repeat the proc- 



104 ARRANGEMENT OF PARTICULAR EXCHANGE- VALUES 

ess, and keep on doing so till the last result tallies, or nearly 
tallies, with the preceding. But there is no certainty in this 
very laborious method. 

A more theoretical objection, however, exists. This is di- 
rected against the very foundation upon which the position rests. 
The position involves the conception that we are measuring the 
variation in purchasing power of a certain total sum of money 
(or of sums of money having the same general exchange-value) 
over a certain class of commodities, and the variation in the pur- 
chasing power of another certain sum of money over another 
class, and so on, and are seeking to average these variations. 
But this does not correctly represent what we are seeking. It 
represents a method of constructing a sunilar world at both 
periods like the one which in the preceding Chapter itself was 
subordinated to another method. What we are seekino; is to 
average the variations in the exchange-value of one given total 
sum of money in relation to the several classes of goods, to 
which several variations must be assigned weights proportional 
to the relative sizes of the classes. Hence the relative sizes of 
the classes at both the periods must be considered. And now, 
if for instance at the first period one class be three times as large 
as another, and at the second the latter be twice as large as the 
former, it is impossible to say which are the relative sizes that 
are common to both the periods. We are left with the position 
requiring a mean between the relative sizes at both periods. 
And the question arises, what kind of a mean is this to be ? 

§ 6. At first sight it might be thought sufficient to add up the 
weights of every class at the two periods and to divide by two. 
This w^ould give the (arithmetic) mean size of every class over 
the two periods together. But such an operation is manifestly 
wrong. In the first place, the sizes of the classes at each period 
are reckoned in the money of the period, and if it happens that 
the exchange-value of money has fallen, or prices in general 
have risen, greater influence upon the result would be given to 
the weighting of the second period ; or if prices in general have 
fallen, greater influence would be given to the weighting of the 
first period. Or in a comparison between two countries greater 



THE QUESTION OF PERIODS 105 

influence would be given to the weighting of the country with 
the higher level of prices. But it is plain that the one period, 
or the one country, is as important, in our comparison between 
them, as the other, and the iveighting in the averaging of their 
weights should really be even. To employ here the same sort of 
correction by approach as suggested in the similar case preceding 
would not only be extremely laborious, but Avould be useless be- 
cause of the next reason. In the second place, even if the ex- 
change-value of money happens to remain stable, or to have 
varied so slightly as not to be appreciable in this way, as may 
often occur, this operation can be justified only on the supposi- 
tion that the problem is that of putting together the economic 
individuals of the first period and the economic individuals of 
the second, and of then seeing how the combined quantities re- 
late to one another. But this is not the problem, because, by 
the hypothesis, the two periods are distinct, and, while they can 
be compared, they cannot be united. Our only data are prop- 
erly the relations that the class A is so many times more or less 
important, or large, than the class B at the first period, and at 
the second so many other times more or less important, or large. 

Nor is the simple arithmetic mean of these relations, by add- 
ing them together and dividing by two, the proper one to 
draw. The proper mean in cases of thife sort is the geometric. 
Of course, as above remarked, we must employ even weighting 
in drawing this mean. 

The reason why the arithmetic mean is improper and the 
geometric mean is proper is a purely mathematical one, which 
will be examined more fully in a later Chapter. In general, the 
operation of adding the separate terms in ratios for the purpose 
of getting the average ratio is unallowable, although it may be 
correct in special instances. The operations to be performed 
with ratios are principally division and multiplication. In par- 
ticular, the true position in the case before us may be shown as 
follows. Let the class A at the first period be r^ times as large 
as the class B and s^ times as large as the class C. Then at this 

period the class B is - times as large as the class C. And at 



106 ARRAXGEMENT OF PARTICULAR EXCHANGE- VALUES 

the second period let the classs A be r^ tunes as large as the class 
B and s.-^ times as large as the class C. Then at this period 

the class B is — times as laroe as the class C. IS^ow if the arith- 

metic mean be used in averaging these relations in order to get 
the size relationship for both the periods together, we should have 
to say that A is l(i\ + r.^) times as large as B and J(s^ + s^) 
times as large as C. Then if we draw the relationship between 
B and C from these relations, we should have to say that B is 

-^ ^ times as large as C. But if we draw the relationship 

between B and C in the same way from the first separately 
given relations between B and C, we should have to say that B 

is 2 1 ^ + " ) times as large as C. But this expression is not 
the same as the preceding, and is equal to it only m special cir- 
cumstances I namelv if r, ==■ r„, or if — =— |. Therefore the 

employment of the arithmetic mean involves an inconsistency, 

which shows it to be absurd. The o-eometric mean is free 

from this inconsistency. This mean being employed, the size 

relationships for both the periods together are to be expressed 

by saying that A is V)\ ■ i\ times as large as B and i/s^ • s^ 

times as large as C, wherefore, using these relations, we must 

-\/ g . g 
say that B is . ^ - times as large as C. And if we used the 

first separately given relations between B and C, we should have 

Jo Q 

— . — times as large as C ; ^\^hich expression 
* 1 ''2 
diifers from the preceding only in form, and is universally equal 

to it. 

With many classes, this operation of geometrically averaging 

the numbers of times each is more important than another would 

be interminable. It can be simplified. All we have to do is 

to draw the geometric mean of the loeights of every class at the two 
periods, and to take the results as the weights of the classes for 
both the periods together. The relative weights so obtained are 



THE QUESTION OF PERIODS 107 

the same as if we compared every class with every other in the 
way described. This may be shown as follows. The total mass- 
qnantity of the class A, which is supposed to be the same at both 
the periods, may be represented by x, and the total mass-quantity 
of the class B by y (.v and y standing for certain numbers of cer- 
tain mass-units of A and B, what these mass-units are l)eing in- 
different). And the price (of the mass-unit before used for the 
quantity) of A may be represented by a^ at the first period and 
by a.^ at the second ; and the ])rice (of the mass-unit before used 
for the quantity) of B may be represented by /3j at the first 
period and by /9., at the second. Then the weight of the class A 
is xa^ for the first period, and for the second xa.^ ; and the weight 
of the class B is y^^ for the first period, and for the second y^^. 
Let xa^ = r^ ■ y^^, and xa^ = r^ • y^^. Then by the above method, 
the weight of the class A for both the periods is vi\ ■ r^ times 
the weight of the class B. Now the geometric mean of the 
weights at both periods of the class B is I'^yj^i • yi^2 — V^ ^1^2 j 
and that of the two weights of the class A is V i\y^^ '^'^^2 = 
//I /9j/9', /-^r,,. The relationship between these is 

Av^eight of A for both periods VV ^^^.-^ • i\r.^ , 

weight of B for both periods yi^Wl^, ^ ^ ' 

the same as before. What is here demonstrated of two classes, 
may be demonstrated in the same way of three classes, of four 
classes, and of any number of classes. 

It is plain also that this method is unaffected by the possible 
variations in the exchange-value of money, or the general level 
of prices. For, statically considered, the variation of the ex- 
change-value of money affects all prices alike. Then, at the 
second period, if the prices of the mass-unit of A is ta^ instead 
of «.„ the price of the mass-unit of B will be tj^.^ instead of j^.^, 
and so on. This t will appear as V't in the weight of every 
class for both the periods together ; and so will not alter 
the relationship between these weights, disappearing in the 
comparisons just as V l^^^.^ ^^^ done. Hence even weighting is 
here really carried out in our averaging of the weights of every 
class at both periods, or in two countries. 



108 ARRANGEMENT OF PARTICULAR EXCHANGE-VALUES 

§ 7. We may illustrate this position by an example, which 
will not only render it clearer, but will als^j disclose certain im- 
portant details. Suppose at both periods we have seven pounds 
of A and seven pounds of B (or any greater quantities in the 
same proportion), and that at the first period each of these seven 
pounds is valued at 100 dollars, but that at the second period 
the money -value of the seven pounds of A is 196 dollars, and 
the money-value of the seven pounds of B is 4 dollars. (Such 
extreme variations are purposely chosen, because they bring out 
the distinction more clearly.) Then, for both the periods to- 



gether the class A is 



19600 ^ ^. T ^1 ^1 1 -D 

/ tunes larger than the class B, 



S 400 



— which relationship is also sho^^Ti by the fact that I'' 19600 = 

^ 140 

140 and l' 400= 20, and —-=7. Then it is proper to say 

that over these two periods together the class A contains seven 
economic individuals, and the class B one economic individual ; 
which means that the economic individual in A consists of one 
pound of A and the economic individual in B consists of seven 
pounds of B. Now the money-value, or price, of the eco- 
nomic individual in A, namely one pound of A, is $14.2857 
at the first period, and^tne second §28.00, while the money- 
value, or price, of the economic individual in B, namely seven 
pounds of B, is $100.00 at the first period, and at the second 
$4.00. Thus the individual in A is at the first period one 
seventh as -large, and at the second seven times as large, as the 

individual in B ; for 14.2857 =—^ , and 28 = 4 x 7. And the 

geometric mean price of these individuals at both the periods 
is the same, being in each case $20.00; for 1 14.2857x28 
= t/100x4 = 20. 

These relations are imiversal, as may be demonstrated in the 
following simple manner. Using the same symbols as above, 
the total money-values of the class A are xa^ at the first period 
and xa.^ at the second, and those of the class B yj3^ and yj3^ at 

the two periods respectively. Therefore, the class A is - ^l^p-^ 



THE QUESTION OF PERIODS 109 

times larger (or smaller) than the class B, or, in other words, if 
the class B be regarded as containing one economic individual, 
being taken, so to speak, as the unit-class, the class B contains 

- ^j-.^ individuals. Then the money- value, or price, of the in- 

1/ ^ P1P2 

dividual in the class A is — r^=^ = ^ .7/ k /9,/9., at the first 

y ^^A _ 

xa^ \^ 

period, and at the second 7^^ = Ji . ^ V 3,3., ; while the 

money- value, or price, of the individual in the class B is at the 
first period y3^ and at the second yj9^. Now \~~ • y l-^^J^^ : 

?l3i '•'• 1/1^2 ' \\ ^ ' y '^'^ i^xi^-i'y ^"^^ *^^^ geometric mean of the 

prices of the individual in A at both the periods and the geo- 
metric mean of the prices of the individual in B at both the 
periods are both y V j^J^^- What is so demonstrated of two 
classes, can be demonstrated of any number of classes. 

A consequence of these relations is that if we suppose the 
prices of these individuals in A and in B, etc., to vary at a uni- 
form rate (that is, at the same percentage, compounding), their 
prices will pass the geometric mean, the same in all cases, at the 
same moment, which is the timal half-way point, and they will 
then be actually of the same size or importance (in exchange- 
value), and at any other moment their sizes will be above and 
below this in corresponding proportions. These, however, are 
imaginary or ideal relationships, as we cannot expect prices to 
vary uniformly, and if the periods be contiguous, the price-vari- 
ations are really within each period, and not between them. 
Still they are the relationships we should demand when we con- 
ceive of the problem in its ideal conditions. 

We have, then, four reasons for preferring the geometric 
mean between the total money-values at the two periods, 
namely, to recapitulate : (1) the geometric mean avoids an in- 



110 ARRANGEMENT OF PARTICULAR EXCHANGE-VALUES 

consistency into which the arithmetic mean falls.; (2) it alone ^^ 
is unaffected by variations or differences in the exchange-value 
of money ; (3) according to it alone the economic^individuals in 
the classes have inverted relationships at each of the periods, 
and (4) coincide in size or importance at the timal half-way 
point, if their price-variations are supposed to be at a uni- 
form rate. 

So conceived, the economic individual in every class, when 
the mass-quantities are constant over both periods, is composed 
at both periods of a constant mass-quantits^, which varies in im- 
portance or economic size (exchange-value), in such a way that 
at the one period it is as much more or less important than every 
other as at the other period it is less or more important than 
every 'other, so that altogether it has the same importance as 
every other. Naturally the sizes of the classes are according to 
the numbers of such economic individuals they contain. 

§ 8. III. Lastly we have to consider the cases, probably the 
only actual ones, when both the mass-quantities and the total 
exchange-values of the classes vary from the one period to the 
other. Here there are only three, or at most four, main posi- 
tions that claim our serious attention. The first is that we 
should take the smaller total exchange-value of every class at 
either period as its Aveight in the comparison of the tjvo periods. 
The second is like unto this, from the point of view of the mass- 
quantities, namely that we should take only the smaller mass- 
quantity of every class at either period, and then treat these 
quantities as we would treat them were they the only ones at 
both the periods, taking for the weight of every class the geo- 
metric mean between the total money-values of these reduced 
mass-quantities at the prices of each of the periods. The third 
is that we should treat the total money- values of the total mass- 
quantities at both periods in the same way we have just recom- 
mended for treating the total money-values of the mass-quan- 
tities that are constant over both periods, namely by taking for 

13 Consideration of the harmonic mean is here omitted because no claim has 
ever been put forward for it, and no reason is apparent in its favor. It involves 
the same inconsistencies and difficulties, inverted, as the arithmetic. 



THE QUESTION OF PERIODS 1 1 1 

the woio^ht of every elass the geometric mean between the class's 
full total money-values at both periods. Among the several 
suggestions above reviewed for treating the divergent mass- 
quantities only one has any good claim for consideration. We 
cannot use the mass-quantities of each period separately, in the 
system of double 'weighting as employed by Drobisch, because 
that perverts the problem, as we shall find in the next chapter. 
There may be other systems of double weighting, as notably one 
invented by Professor Lehr ; but we may more conveniently 
examine these later. And at all events, so long as we consider 
the question of averaging price-variations, instead of measuring 
variations of price-averages — a distinction which will be made 
plainer in the next Chapter, — we shall need to choose a method 
of single weighting ; and the single system itself that is adopted 
will be the basis for any double weighting we may later find 
proper to adopt. To take the mass-quantities of either period 
alone is absurd, as neither is of more importance than the other. 
To make one calculation on the mass-quantities of the one period 
and another on those of the other, and then to take the (arith- 
metic) mean between the two results, is an arbitrary proceeding. 
The only position hitherto recommended that deserves any at- 
tention is that of taking the (arithmetic) means between the 
mass-quantities at the two periods, or, which is the same thing, 
the sums of the mass-quantities at both the periods, and treating 
these as if they were mass-quantities constant over both the 
periods.'* And yet a question will arise as to whether the geo- 
metric mean is not better here also. 

The first of these methods is very much like the method above 
discussed of taking the smaller total money- value at either period 
when the mass-quantities are the same at both the periods. It 
shares the defects of that method, and adds to theA another. 
This is that if we happen to be comparing a prosperous period 
with a dull period, or a large country with a small country, the 
smaller total money-values may be mostly, and even all, those 
of the dull period, or of the small country, even though the ex- 

^4 This and the preceding method probably yield very nearly the same results. 
There is one condition in which their results always agree exactly. This is in- 
dicated in Appendix C IV J 2 (3). 



112 ARRAXGEMEXT OF PARTICULAR EXCHANGE-VALUES 

cbange-value of money may be tlie same at both the periods, or 
in both the countries, and the total mouey-vakies correctly repre- 
sent the total exchange-vahies. For this added defect, however, 
a correction is possible, which will be pointed out presently. 

The second has more to recommend it. We are seeking to 
measure the exchange-value of money hi relation to other classes 
of things, all which compose a whole. If all these classes of 
things consist of the same masses at both periods, the material 
whole, or world, which they compose, is the same, or similar, at 
both periods. And that the material world should be the same, 
or similar, at both periods, in spite of, and underneath, the vari- 
ations in the exchange-values of the things in it, has the ap- 
pearance of being a postulate of simple mensuration. Now by 
eliminating all the surplus mass-quantities that exist at either 
period alone, we pick out all those mass-quantities which are 
common to both the periods, so that a material world composed 
of these mass-quantities is actually the same, or similar, at both 
periods. ^^ At all events the selection of these mass-quantities 
seems to provide an answer to the objection advanced by Professor 
Sidgwick and by Dr. Wicksell. There is also another argument, 
apparently in favor of this position, that may be derived from a 
more general position, which here deserves to be noticed. 

§ 9. When new classes of commodities appear upon the scene 
in the course of our comjiarisons, the appearance of these in no 
wise affects the exchange- value of the others. At first, to take 
a simple example, our money-unit might have purchased so much 
of one kind of thing, or so much of another ; that it should 
later be able to purchase the same amounts of these things and 
so much of a new kind of thing, widens the range of its ex- 
changes, but does not increase its power over other thiugs. The 
change is extensive, not intensive. Or, viewed in another way, 
at first the money-unit could purchase half as much of the first 

15 If this be the right position, then the treatment above recommended of the 
special cases when tlie total money-values of every class happen to remain con- 
stant, or to vary in the same proportion, has to be abandoned. For then, some of 
the prices being supposed to have varied, the total mass-quantities must have 
varied, and therefore, according to the present method, only the smaller of 
these total mass-quantities, and their reduced total moaey-values, must be taken 
into consideration. 



THE QUESTION OF PERIODS 113 

and lialf as much of the second kind of thing-.s ; and later it will 
purciia.se a third of the first and a third of the second and a third 
of the new article. Here what is gained in extension, is lost in 
intension. A new class of goods adds a new particular exchange- 
value to the general exchange- value of everything else ; but this 
new exchange-value is only equal to each of the old ones, and 
equal to the general exchange-value (according to Proposition 
VII.), and therefore makes no change in any of those magni- 
tudes. A variation in anything's general exchange-value can be 
brought about only by a variation in one or more of its already 
existent particular exchange-values ; and such a variation can 
take place only relatively to a class of objects existing at both 
the periods. A particular exchange- value relatively to a class 
which exists only at one of the periods compared cannot have 
varied. Consequently a class of things is not to be counted in 
the one period ivhen it is not counted in the other .^'^ And in a 
series of comparisons, if a new class reaches sufficient importance 
to deserve to be counted at a certain period, in comparing this 
period with the preceding, in which it is not counted, it must 
still be neglected ; and it is to be counted only in the comparison 
of the next period with this jjeriod, and thereafter.^'' And in 
comparing with the present a period in the distant past we must 
leave out the classes which have come into existence since and 
those which have passed out of existence, comparing only those 
parts of each world which are common to both. Here, however, 
there is much necessary imperfection in the comparison, because 
some classes may have passed out of existence only recently, and 
others may have come into existence soon after the first period, 
and these may have varied in the intervening periods. ^^ 

1* Thus Fauveau said he assumed only the same species of things at both the 
periods, B. 54, p. 355. — This principle may be violated in Drobisch's method. In- 
diiFerence to its violation is shown by Lindsay, B. 114, p. 26. The need of ob- 
serving it is treated as a defect by Nicholson, B. 94, if. 313-315. 

^■'Attention was called to this problem by Sidgwick, who thought it one of 
great difficulty, B. 56, p. 68. It was solved by Marshall, B. 93, p. 373 n. The 
solution was adopted by the British Association Committee, First Report, B. 99, 
p. 250. 

^* Hence in such cases true comparisons can be made only bypassing through 
all the intervening periods. Where this is impossible, a true comparison is im- 
possible. 



114 ARRANGEMENT OF PARTICULAR EXCHANGE- VALUES 

The knowledge gained by this investigation into the want of 
influence of new classes it may be well now to state in proposi- 
tioual form. From this investigation (and really as a corollary 
to Proposition VII.) we perceive that the height of the general 
exchange-value of anything is in no loise determined hy the num- 
ber of its particular exchange-values, but, solely by their (common^ 
height (Proposition XLI.). And so, there being no variation of 
exchange-values (or of prices), the appearance of a new class of 
things, with any exchange-value whatsoever, has no influence upon 
the height of the exchange-value of anything already existing, nor 
can it have any such influence until it has existed over at least 
two periods, thus having time to remain constant or to vary ; 
nor can the disappearance of an old class have any such influence, 
no matter lohat loas its exchange-value at the time when it disap- 
peared (Proposition XLII.). 

Now what is thus shown to be true of the appearance or dis- 
appearance of a class, it might be argued, is also true of the ap- 
pearance or disappearance of individual (physical or material) 
things in old classes. It might be argued that just as we must 
count only the classes of things which are common to both the 
worlds compared, so in each class we must count only the (ma- 
terial) individuals which are common to them both. The reason 
why the classes which appear only at one of the periods cannot 
be counted is that such classes have neither varied nor remained 
constant in exchange-value and money has neither varied nor 
remained constant in exchange-value in them. The same reason 
might seem to apply to (material) individuals which are present at 
only one of the periods. These have neither varied nor remained 
constant in exchange- value, nor has money either varied or re- 
mained constant in exchange-value in them. Therefore these 
single-period individuals, it might seem, should be neglected in 
our calculations. 

§ 10. What has above been said of classes is true of (material) 
individuals as follows. The height of the general exchange- 
value of anything is not determined by the number of its in- 
dividual exchange-values, but solely by their (common) height 
(Proposition XLIII.). Hence, if there are no variations of 



THE QUESTION OF PERIODS 115 

prices, or of the exchange-values of things in money, and of 
money's inverse exchange-vahies in all other things, so that the 
general exchange-value of money is constant (according to Prop- 
osition XXVII.), then the coming into existence of new indi- 
viduals, or the passing out of existence of old ones, with the 
constant prices of the other individuals in the same classes, does 
not affect the height of any particular exchange-value of money, 
and consequently not the height of its general exchange-value.^^ 
In other words, a change in the size of any classes, without a 
variation in any exchange-values (or prices), does not cause a 
variation in anything's general exchange-value (Proposition 
XLIV.)."" This means that wJien there are no variations of ex- 
change-values (or of prices), the weighting of the classes is indif- 
ferent. Hence it is indifferent, in this case, whether we neglect 
the surplus individuals or not. It may be added that also if all 
prices vary in the same proportion, changes in the sizes of any 
classes do not affect the variation in the exchange-value of money, 
which is the inverse of this common pjr ice-variation (Proposition 
XLV.)f ^ and consequently the weighting of the classes is indif- 
ferent also in this case. 

But when there is irregular variation of some exchange-values, 
or prices, between two periods, then the weighting is very im- 
portant. And now also it is important whether any of the 
classes ^^ has augmented or diminished or remained constant in 
size. That allowance should be made for such changes in total 
exchange-value (or in total money-value) size, we have already 

^* For at a given moment final utilities are, to the people at large, according 
to prices. At a given moment, then, prices being given, if people spend their 
money more on one- class than on another, or reversely, this only shows that they 
are choosing the tinal utillties.in a different manner, not that they are getting 
more final utility by spending their money in one way than in another. There- 
fore at two moments, or periods, all prices still being the same, if people actually 
do spend their money differently, this shows nothing more or less than was shown 
in the previous supposition. 

-" This Proposition flows directly from Proposition XXVII., which is uncon- 
ditional. 

-i^This Proposition flows directly from Proposition XVII., which is uncon- 
ditional. Compare also Propositions XXXV. and XXXVI., from which a simi- 
lar corollary may be drawn concerning the weighting, in all cases, of any class 
whose price varies inversely to the exchange-value of money — either in all things 
or in all other things, according to the measurement that is being made. 

"2 Except as indicated in the preceding note. - 



116 ARRANGEMENT OF PARTICULAR EXCHANGE-VALUES 

conceded. Then why should not allowance be made also for 
changes in material size ? The economic individual may change 
in exchange-value size between the two periods. Then why 
should not the economic individual change also in material size? 
It is true that it is only the lesser mass-quantity of any class 
existing at either period, or the mass-quantity common to both 
periods, that has varied or remained constant in exchange- value. 
Yet money has varied or remained constant in exchange- value 
in a changing mass-quantity of this class, just as it has varied 
or remained constant in exchange-value m a changing exchange- 
value quantity. When a class is wholly absent from the one 
period, though present at the other, we have no price-quotation 
of it at all at the one period, and so it can have no variation or 
constancy in its price or exchange-value.^^ But when merely a 
certain material quantity of a class is absent from the one period, 
yet, as its similar mates are present at both periods, we do know 
the variation or constancy of money in exchange-value in this 
changing class of things. 

§ 11. A practical objection also exists against taking only 
the lesser mass-quantities at either period, like that brought 
against taking only the lesser total money-values of the classes 
at either period. This is that if we are comparing a prosperous 
period with a dull period, or a large country with a small 
country, the mass-quantities might be only those of the small 
country — ^which would be treating the small country as if its 
conunodities were more important than those of the large country, 
— or only those of the dull period, which might in one case be 
the earlier, and in another the later period, so that in the former 
case we should be using the mass-quantities of the first period in 
the comparison of the two periods, and in the later the mass- 
quantities of the second period, the only reason for the difference 
in the choice of the period being that more importance is attached 
to the conditions in the dull period than to those in the pros- 
perous period, — which is absurd. 

Still a correction of this defect in this method — and at the 

2 3 For surely we cannot supply what would have been its pi'ice had it existed, 
as suggested by Nicholson, B. 94, p. 314. 



THE QUESTK^N OF TERIODS 117 

same time of the similar defect in the first-noticed method — 
could be made in the following manner. Finding the total 
money-expenditure at each of the periods, or in each of the 
countries, reduce the expenditure in the one to the expenditure 
in the other, and reduce all the mass-quantities in the same pro- 
portion ; then take the smaller of these mass-quantities at cither 
period, or country, and operate as before — or, in the other 
method, take the smaller of the total money-values so reduced. 
This correction, however, generally needs a further correction. 
For it involves that the total money-expenditures so compared 
shall be of the same exchange-value ; and, therefore, it can be 
safely employed only when the exchange-value of money is the 
same at both the periods, or in both the countries. When this 
is not the case, or not known to be the case, as always when we 
are attempting to measure the exchange-value of money, the 
only further correction can be by an uncertain method of aj)- 
proach, like that previously noticed. For perhaps the truth 
may be reached by taking the first result of this method as 
representing the variation in the exchange-value of money, and 
then again reducmg the total money-expenditures to the same 
total exchange-value expenditure on the assumption of this 
variation being correct, and repeating the operation until the 
result tallies with the last assumption. 

Another of the above suggested methods, namely that of tak- 
ing the arithmetic mean between the mass-quantities at each 
period, or, which is the same thing, the total mass-quantities 
over both the periods, has no theoretical reason in its favor ; for 
it is not the total mass-quantities during both the periods, or the 
halves of these, that have varied from the one period to the 
other, since more than half of them may be within one of the 
periods. 

Moreover this method stands in need of the same sort of cor- 
rection as the preceding. For between a small country and a 
large country, or between a dull period and a prosperous period, 
the arithmetic mean of the mass-quantities would be more in- 
fluenced by the conditions in the large country, or in the pros- 
perous period — the influence here being on the (»p])osite side 



118 ARRANGEMENT OF PARTICULAR EXCHANGE-VALUES 

from what it was on in tlie preceding cases. The correction is to 
be made in the same way as before, namely by reducing the mass- 
quantities, before averaging them, in the same proportion, so 
that their grand total exchange-values — not merely money- 
values — shall be the same at each period ; all which again in- 
volves the need of approach through many assumptions and 
repetitions, so that, from being a simple and convenient method, 
it becomes a very laborious one. 

A simpler means of avoiding at once the difficulty arising 
from differences in sizes of the countries or in prosperity of the 
periods and that arising from variation in the exchange-value 
of money, consists in eschewing the arithmetic mean of the mass- 
quantities and substituting the geometric. 

It may be, however, that in spite of its theoretical insufficiency, 
the method of taking merely the arithmetic means of the mass- 
quantities, uncorrected, may in ordinary cases yield results so 
near the truth that it may be preferable to any of the other 
methods thus far examined, and on account of its greater con- 
venience may be even preferred to the truer methods which we 
shall later discover. But the practical merits of these methods 
it is impossible to examine here. We shall examine them 
later in connection with the averages that are to be drawn of 
the price variations. 

§ 13. The suggested method above set down as the third still 
remains for examination. This is simply to draw the geometric 
mean of the full weights of every class at both periods, that is, 
of the total money-values (which indicate the relative total ex- 
change-values) of the classes at both the periods, without having 
any more concern for the changes in the mass-quantities of the 
classes than for the changes in their total exchange-values. In 
the very first case considered, where the total money-value of 
every class happens to be constant over both periods, or to change 
in the same proportion, so that the relative sizes of the classes 
remain unchanged, we have seen that the most natural treatment 
is to take these relative money-values as the weights of the 
classes. Then, in the simplest cases, the economic individual is 
a constant money- value at both periods, but is a changing mass- 



THE QUESTION OF PERIODS 119 

quantity. lu the second general case, where the mass-quantities 
in all the classes happen to be constant over both periods, the 
economic individual we have unhesitatingly found to be a con- 
stant mass-quantity at both periods, but a changing money -value 
(or exchange- value). Neither of these cases is likely ever to 
occur in actuality. Now, when both the total money-values and 
the total mass-quantities in every class are various at both the 
periods, it seems most natural, and not improper, to combine 
both these conceptions, and to liave for our economic individual 
one changing both in money-value (or exchange-value) and in 
mass-quantity.^^ As before, the nature of this individual may 
be best understood by aid of an example. 

Suppose at the first period we have 100 pounds of A at $1.00 
apiece and 100 pounds of B at $1.00 apiece, and at the second 
period 50 pounds of A at $1.96 apiece and 1,000 pounds of B 
at $0.04 apiece. The total money-values of the class A are 
$100 at the first period and $98 at the second, and those of the 
class B $100 and $40 respectively. The relative sizes of these 
classes, therefore, according to this method of measurement, are 
1/9800 = 98.995 for the class A, and l/4000 = 63.246 for the 
class B. Or if the latter be taken as one unit, the weight for A 

. 98.995 -,,../' (9800 ,/K~rn\ rp, . ,i • 

IS = 1.5651 = ^ = 1/2.45 ) • Thereiore, the m- 

63.246 \ \4000 / ' 

dividual in the class A consists at the first period of :r~F-^'^ = 

^ 1.56o 

50 
63.89 pounds, worth $63.89, and at the second of -^-^- = 
^ TT 7 1.565 

31.945 pounds, worth $62.61 ; and the individual in the class 

B consists at the first period of 100 pounds, worth $100, and at 

. n n 1 ^ T. 100 62.61 

the second of 1000 ijouuds, worth $40. But >jiS"^7i = — ttt' 

= 1.565, that is, the individual in B is 1.565 times more valu- 
able than the individual in A at the first period, and at the second 
period the individual in A is 1.565 times more valuable than the 
individual in B, although the individual in A has contracted by 

24 If this method be admitted, both those methods are to be retained, and em- 
ployed when the conditions are met. This method is comijrehensive, and includes 
both those. 



120 AEEANGEMENT OP PAETICULAE EXCHANGE- VALUES 

half, and the individual in B has grown tenfold, in mass-size. 
Thus in spite of the changes in their mass-sizes the individuals, 
so obtained, in the classes have inverse relations of money-value 
(and consequently of exchange- value) at each of the two periods, 
and therefore are equally valuable, or equivalent, over the two 
periods together. Also l/63.89 x 62.61 = l/lOO x 40 = 63.- 
246, that is, the geometric mean of the prices of the individual 
in A is the same as the geometric mean of the prices of the in- 
dividual in B, which means that at the half-way moment between 
the two periods, the prices being supposed to vary at a uniform 
rate, the prices of the tsvo individuals are the same, so that then 
these two individuals are actually equivalent. 

That these two relationships are universal, can be demon- 
strated as easily as before in the case when the mass-quantities 
were the same at both periods (see above, § 7). We have 
only to distinguish x into x^ and x^, and y into y^ and y^, and 

y^y^y^il^^ ^^ *^® ^^'^^ period and ^f"-^^ • '^ yiyj^il^2 ^^ ^^^ ^^^~ 

ond, and everything works out as before. 

Variations in the exchange-value of money have no influence 
to derange the weights here, just as they have none in the cases 
where the mass-quantities are the same at both periods, every 
weight being afPected in the same proportion here as there. 
Nor does a difference in the sizes of the countries, or in the 
prosperity of the periods, have any such deranging influence. 
The geometric mean with even weighting being employed, equal 
weight is really attached to the weights of each period. There 
is no need of correction. This method of weighting, then, is 
not only the best, but even the simplest. 

At all events this method seems to give us the true concep- 
tion of the economic individual in a comparison between two 
periods. That the variations of the prices of the classes, be- 
tween two periods, should be weighted according to the relative 
numbers of such individuals the classes contain over the two 
periods, would seem to be proper. Or if we average prices at 



WAGES EXCLUDED 121 

each period separately, preparatory to comparing the averages, 
it would seem proper to make use of the numbers of these indi- 
viduals the classes contain at each period, determined with refer- 
ence to the other period with which it is compared. 

§ 14. The subject has to be left in this somewhat unsatis- 
factory state until we have examined the question of the aver- 
ages in which these systems of weighting are to be employed. 
One system of weighting may jierhaps be found suitable for one 
kind of averaging, and not for another. And one kind of 
weighting with one kind of averaging may be found to yield 
exactly, or very nearly, the same results as another kind of 
weighting with another kind of averaging. Hence our choice 
may be more narrowed than it is at present. 

So far we have been attempting, without complete success as 
yet, to reach the theoretically exact position. To attain to this 
position itself would, however, be of little service, unless we 
have very exact data to apply the theoretically correct method 
to. Hence the theoretically correct method of weighting will 
be serviceable only for measuring the course of the exchange- 
value of money during present time, as we advance into the 
future. For reviewing what is already past, it is hopeless to 
expect to find data sufficient to justify us in making use of any 
but a rough and ready weighting, the same for many consecutive 
periods together, the finer shades of difference from year to year 
being untraceable. This weighting, the same for many periods, 
must either be some general average of the relative importance 
of the classes over these periods, or be according to some gen- 
eral average of the mass-quantities in the classes over these 
periods. Which of these methods is the better, and of what 
sort the average ought to be, we must postpone examining till 
we have reviewed the subject of weighting in connection with 
the subject of averaging the price variations. 

VI. 

§ 1 . One more item remains, about which there has been dis- 
pute. Many economists have maintained that among the prices 
of thiuffs which are to be counted in measurino; the exchange- 



122 SELECTION OF PARTICULAR EXCHANGE- VALUES 

value of money we should include also the " price of labor/' 
namely, wages and salaries.^ 
> No opinion could be more erroneous. 

§ 2. In the first place labor has no exchange- value. Labor 
is not a possessible thing : it does not pass from one owner to an- 
other ; it is not exchanged for anything else. What ! it may be 
said, does not the employer pay money for labor, and does not 
the laborer get money for his labor ? By no means. The es- 
sence of the contract between a manufacturer and his employees 
is that the former shall put materials and machines in the hands 
of the latter and shall take the products which they shall make 
— ^he buys from them the improvements they make, they sell to 
him those improvements.^ Even in domestic service what the 
employer pays for is the product of labor : — the charwoman is 

^ "Wages were included in the lists, with the prices of commodities, for calcu- 
lating variations in the " value " of money, by Dutot, Evelyn and Young. Adam 
Smith found the " value " of silver at different epochs by considering the sums 
needed to purchase "the same quantity of labor and commodities," op. cit., p. 
101. (His inexactness is shown by his making the same measurement by " the 
quantity of labor which any particular quantity of them [gold and silver] can 
purchase or command, or the quantity of other goods which it Avill exchange for," 
p. 14.) McCuUoch : to be constant in " exchangeable value" a thing must " at 
all times exchange for, or purchase, the same quantity of all other commodities 
and labor," Political economy, p. 213 (and in Note to Wealth of nations, p. 439). 
Roscher gives a truly German reason for assigning "an important place " in the 
lists to daily wages : " The desire to exert influence upon other men and to be 
prominent socially is a very universal one ; and of its attainability there is no 
better sign than the power of disposal over many days of labor," B. 32, § 129. 
That wages should be counted along with commodities has been of late asserted 
by Martin, op. cit., p. 626 ; Nasse, Die Wdhrungsfrage in Deutschland, Preus- 
sische Jahrbiicher, 1885, p. 313; Giffen, B. 44, pp. 127, 128, B. 45, q. 780; New- 
comb, B. 76, p. 212 ; Wasserab, B. 105, p. 75 ; G. P. Osborne, Principles of econ- 
omics, Cincinnati 1893, p. 332 ; Lindsay, B. 114, p. 332 ; G. H. Dick, International 
bullion money, London 1894, p. 3 ; Wiebe, B. 124, pp. 168-169 ; Edgeworth, B. 
65, p. 386 ; A. M. Hyde, Gold, labor and commodities as standards of value, 
Journal of Political Economy, Chicago, Dec. 1897, p. 97; Parsons, B. 136, pp. IV., 
97, 115, 128. 

2 The dif&culty in carrying out the ordinary opinion is well shown by James 
'SUllin his Elements of political economy (2d ed., 1824). He starts out by say- 
ing ; " The laborer who receives wages sells his labor. . . . The manufacturer who 
pays these wages buys the labor," p. 21 ; and later says that when t| capitalist pro- 
vides raw materials and tools, and the laborer works up the product, this "be- 
longs to the laborer and capitalist together," but that, "when the share of the 
commodity which belongs to the laborer has been all received in the shape of 
wages [paid in advance of the sale of the product], the commodity itself belongs 
to the capitalist, he having in reality, bought the share of the laborer and paid 
for it in advance," pp. 40-41. On p. 90 he makes both these statements ! 



WAGES EXCLUDED 123 

paid for clean windows and floors made out of dirty windows 
and floors, the cook for cooked food in the place of raw food, the 
waiter for food produced on the table from food produced in the 
kitchen, the coachman for a movirig carriage instead of a station- 
ary one.^ Employers do not want labor — they would pay much 
more if they could be served, like Psyche in Cupid's l)ower, with 
the hands of invisible spirits. What they want, and what they 
pay for, is something- which they cannot get without paying some 
one to produce it for them/ And what laborers give in return 
for their hire is not the labor which nobody wants, but the pro- 
ducts of that labor.^ 

§ 3. But this is not the principal reason, although the princi- 
pal reason flows from this. Labor and material things being 
uninterchangeable, the wages of labor and the prices of com- 
modities are categorically distinct. To put them in the same list 
is to try to mix oil and water. Wages belong to another list. 

3 Some writers have admitted that the wages of " productive hibor " (wages 
paid for the production of goods that are to be sold) are not to be counted because 
they are a factor entering into the prices of tlie goods produced, and therefore, 
being already counted in tliose, they would be counted twice ; but claim that the 
wages of so-called "unproductive lalior" (of domestics, etc., whose products are 
not sold) ought to be counted : — Edgeworth, B. 59, p. 266 n. ; Marshall, B. 93, p. 
372 ; Nicholson, B. 94, p. 324 ; tlte Britisli Association Committee, First Report, 
B. 99, p. 249; H. J. Davenport, Outlines of economic theory, New York 1896, p. 
227; Wicksell, B. 139, p. 18. Here a distinction is to be drawn. The wages of 
domestics, etc., contribute only to the retail prices we pay for what we consume. 
Therefore if retail prices are being used in our lists, these wages would properly 
belong there. But retail prices are usually excluded, for reasons already ex- 
plained. Hence the wages even of domestics should be excluded. 

*So Fonda, B. 127, pp. 14-15, and Davenport, op. cit., p. .53. Cf. Aristotle, 
who said we sliould have no need of workmen if shuttles moved themselves, 
Politics, I., 2, 5. — On the other hand, that labor is the only thing we pay for has 
been asserted by W. D. Wilson, First principles of political economy, Phila- 
delphia 1882, p. 100. 

5 The products of labor for whicli we excliange money may be immaterial 
things — dramatic scenes, music, etc. (pleasureable sensations and thoughts), or 
states and conditions (safety, liealth, etc.) — in short services, if by this term be 
meant, not actions, l)ut the effects of actions. We do not pay musicians for play- 
ing, but for music ; nor lawyers for pleading, but for the results of tlieir plead- 
ing, — often paying, however, for a chance of getting what we do not get. Now 
immaterial products ought, theoretically, to l)e counted in the lists — we might 
couut theatrical tickets, fees, etc. The sole reason why they are to be excluded 
is the practical one already, applied to many material products, that their 
qualities are so diverse, the individual products under the same names so various, 
that they do not form lioraogeneous classes. Yet one of them, transportation of 
persons, we have seen to deserve to be counted. 



124 SELECTION OF PARTICULAR EXCHANGE- VALUES 

Labor is the cost at which the majority of mankind procure 
the commodities whose prices go into the list drawn up for 
measuring the exchange-vahie of money. According as is the 
labor which it costs a man to produce a certam thing, so is the 
cost- value of the thing to him. According to the average labor 
which it costs the producing part of mankind to produce things 
of a certain class, so is the average cost-value of that class of 
things in general. The cost-value of a thing is not necessarily 
according to the labor merely of producing the thing physically ; 
for things are carried to places where they are not so produced, 
and the economic production — oifering in the market^ — of them 
there includes the labor of transportatatiou. The majority of 
mankind, however, do not produce but a small part of the things 
they consume, and so are more interested m the cost of procuring 
things, which is according to the cost of producing other things, 
material or immaterial, and the rate at which these can be ex- 
chauffed for those. Here the idea of .ost-value in the narrow 
sense of cost of production proper, passes over into a wider sense 
of cost of acquisition, in which it becomes identical in magnitude 
with esteem-value. If a man gets a dollar a day for what he 
produces, whether in wages or otherwise, then anything the price 
of which is a dollar costs him a day's work. Such a thmg has 
not to him the exchange-value of a day's work ; for he does not 
exchange the day's work for it, and if he does not get it, through 
not workmg for it, he has lost his day's work as well as the 
thing — he has idleness without the thing in place of the thing 
with work, so that if there is any exchange, it is exchange of 
idleness for the thing, and the thing should be said to have the 
exchange-value, not of a day's work, but of a day's idleness. 
Nothing, however, is reached along this line of reasoning. There 
is properly no exchange, but there is cost.*" There is comparison 

8 The conception of Turgot and Adam Smith that labor is the real price we pay 
for what we get is a misuse of terms, as it confounds "price" with "cost." 
Price, in a wide and in itself not very proper sense, is the thing which one man 
gives to another in exchange for what the other gives to him. Cost is what we 
give up, without anybody else getting it. If a man who owns a ton of coal gives 
it to another in exchange for a ton of iron, the ton of coal is, in this wide sense, the 
price of the ton of iron (and the exchange-value of the ton of iron in coal is meas- 
ured by this amount). If a man burns up a ton of coal in smelting a ton of iron. 



WACJES EXCLUDED 125 

of the pleasure in procuring the thing with the displeasure of 
doing the work. This comparison is the essence of esteem- 
value, which permits a certain amount of cost-value to things, 
and no more. This feature in esteem-value shows itself again 
in the comparison of the pleasure of possessing one thing with 
the displeasure of abandoning the possession and use of another, 
or of renouncing possession and use of other things also pro- 
curable in the place of the thing obtained ; which comparison 
regulates the exchange-values of things. 

§ 4, Now it is desirable that we should be able to measure 
the cost-value of things in both the narrow and the wide sense. 
Industrial progress consists in cheapening the cost-value of 
things, and it is well to know whether progress is being made, 
and hoAV rapidly. But the trouble of measuring the cost-value 
of everything separately might be saved by measuring the cost- 
value of money, provided this be its cost-value in the wide 
sense, identical with est<_ /n-value. For if we should find, for 
instance, that this cost-value of money has been constant, then 
we should know that, as the price of anything varies, the cost- 
value, at least in the wide sense, of that thing directly varies ; 
and, if we have measured also the general exchange- value of 
money by means of an average of prices, then, as the average 
price of all things varies, so does the cost-value of all things on 
the average directly vary. Or if we should find that the ex- 
change-value of money in all other things has been constant, 
but that its cost^value has fallen, then we should know tliat a 
steady price of anything means a fall in its cost-value to just that 
extent, a fall of price a still greater flill in cost-value, or a rise 
of price either a lesser fiill in cost-value or a stationary cost- 
value or a rise in cost- value, according to the proportions between 
the fall of money in cost- value and the rise of the price ; and 
we should know that all other things have on the average fallen 
in cost-value to the same extent as money. Or we might find 

the ton of coal is one of the costs in producing the ton of iron. Labor is another 
one of tlie costs of producing the ton of iron ; and as hibor was expended in pro- 
ducing the ton of coal, labor is an ultimate cost of production (and cost-value is 
measured by labor). Labor cannot be the price of anything, because one man 
cannot give it to another. 



126 SELECTION OF PARTICULAR EXCHANGE- VALUES 

that the exchange-vahie and the cost-value of money have both 
changed, the one rising and the other falling, or both together 
rising or falling, only the one more and the other less. There 
are, in fact, all the five typical possibilities we have noticed in 
another case between the variations of two independent quantities. 
But whichever of these changes take place, when we know their 
proportions, it is easy to calculate out all the particular varia- 
tions concerning which knowledge is desired. Therefore the 
measurement of the cost- value of money is also desirable. 

§ 5. Now the cost- value of anything, in the wide sense, as 
determined by the labor cost of acquisition, and identical in 
magnitude with esteem-value, is measured, not by the average 
labor-cost of producing the thing in question, but by the average 
labor-cost of procuring it. In the case of paper money, it is 
evident that we have no interest in the cost of producing it ; 
and even in the case of metallic money, our interest in the cost 
of producing it is small, as regards the subject before us. We 
are interested m the labor-cost of procuring it. Evidently such 
a thing as the general labor-cost of procuring money is to be 
measured by the average labor-cost to the average man of pro- 
curing money. This may be found by finding the total money- 
earnings of a country in a given time, say an hour, and dividing 
it by the number of workers, with allowance for the number of 
hours a day each one works.^ 

Here incidentally it deserves of notice that in measuring the cost 
of procuring money, we should not confine our calculation to the 
consideration of wages, or of wages and salaries. To confine at- 
tention to wages, and especially to the wages of the cheapest and 

2 There was a dispute in the early part of the nineteenth century as to whether 
the measure of " value " is the quantity of labor needed to produce the thing or 
whether it is the quantity of labor the thing will command in exchange. Adam 
Smith had advanced both positions, and sides were taken by his followers, 
Ricardo assuming the former tenet, while Malthus defended the latter, each 
having partisans. Now it is plain that the measure of cost-value proper is the 
average quantity of labor needed to produce the thing ; and it would seem as if 
the measure of cost-value in the wide sense, or esteem-value, may be taken to be 
the average quantity of labor the thing will command. Thus both those positions 
were correct and incorrect. They were correct each of one kind of "value," and 
incorrect of the other kinds of " value." They were both incorrect of exchange- 
value. 



WAGES EXCLUDED 127 

commonest sort of laborers, or to agricultural laborers, as recom- 
mended by some writers,^ would be like confining our attention 
in the attempt to measure the exchange-value of money to the 
price of wheat or other food, as performed by some of the earlier 
workers in this field. Not only all wages and all salaries, but 
all profits (apart from rent and income, if these be already 
counted in the gross income of those who pay them), should be 
included in the examination, and some means of averaging them 
should be found. The common word for wages, salaries, and 
profits, is earnings. The general money-earnings of the hour's 
work of all the working part of the community should be some- 
how found. Perhaps some earnings, as immeasurable in prac- 
tice, would have to be omitted, just as in measuring the ex- 
change-value of money we have to omit the prices of some 
things. But the eifort should be made to include as many as 
possible. 

§ 6. Over against this measurement of the cost- value of 
money, we have curiosity to know the cost-value in the same 
sense, or esteem-value, of all commodities in general. We 
might be tempted to measure this in the same way, by finding 
the total product of a country in a given time and dividing by 
the number of producers, or of consumers. But to this there is 
objection, in that the total product would have to be measured 
all by weight or all by capacity, and when a change takes place 
between two periods a different result would be reached accord- 
ing as the one or the other of these measures were used — a diffi- 
culty we have already noticed in another connection. For this 
lumping together of a total mass product would mix up mate- 
rials of diiferent qualities and fineness, and from period to period 
a change in the mere total mass would disregard possible im- 
provements or retrogressions in the qualities' — a difficulty Me 
shall meet again. Instead, we should have to measure the vari- 

^ Harris, Essay upon money and coins, 1757, Part I., p. 13 ; Malthus, op. cit., 
pp. 96, 112, 116 ; Shadwell, op. cit., pp. 202-203, and in the Journal of the British 
Association for the Advancement of Science, 1883, p. 626 ; T. I. Pollard, Gold 
and silver weighed in the balance : a measure of their value, Calcutta 1886, p. 75. 

* What we want really to find is the total quantity of pleasure procurable 
from our productive activity. But, in equal quantities, coarse goods do not 
yield so much pleasure as fine goods. 



128 SELECTION OF PARTICULAR EXCHANGE- VALUES 

ation in the cost of production of every kind of article separately, 
by finding its total product in a given time and dividing by the 
number of its producers, at the two periods compared ; and then 
somehow, by some method of averagmg never yet investigated, 
combine the variations in the costs of production of the different 
kinds in one variation (or constancy) of the cost of production 
of all things. This is a very complex method. It is the only 
one possible for measuring general cost-value in the narrow and 
proper sense. But for the wider sense, in which cost-value be- 
comes almost identical with esteem-value, another simpler 
method is also possible. 

This is to find the esteem-value of money alone, and to find 
the exchange-value of money alone ; for by comparing the results 
of these two measurements we can get the result desired. We 
thus see that we have need of two distinct operations, which are 
supplementary to each other, and together help us to reach 
a final result in regard to the esteem-value of commodities in 
general. 

§ 7. Now if, instead of these two separate operations, we 
should attempt to perform a single operation by including earn- 
ings in the same measurement with the prices of commodities, 
we should form a hodge-podge that has no meaning. Its results 
would indicate neither the exchange-value nor the esteem-value 
of money, and as it is undertaken only with a view to measur- 
ing the " value " of money, it would mean nothing, there being 
no economic value apart from the four kinds we have analyzed 
out, nor any value compounded of any two or more of these. 
For instance, take the case w^hich is believed to have been going 
on for the past twenty odd years. Prices in general have been 
falling, and at the same time money earnings in general have 
been rising. A whole school of modern writers, reviving views 
which triumphed during a similar period in the early part of 
the century, have concluded from this state of things that the 
" value " of money has been about stationary.^ This result 

° E. g., " Gold prices fell only 19 per cent, from 1873 to 1891. . . . Wages, in 
gold, rose more than 14 per cent, from 1873 to 1891. . . . The advance in wages 
since 1873 so nearly offsets the decline in prices that when fairly tested by both 
prices and wages the value of gold in 1873 and 1891 was practically the same," B. 



AVAGES EXCLUDED 129 

would be obtained, in fact, if we put money earnings, which 
have risen, and the prices of commodities, which have fallen, 
in the same table, and, attaching equal im])ortance or weight 
to each, drew an average between them all — on the supposition 
that the rise of the former has been about equal to the fall of 
the latter. And yet it is })lain that the fall of the prices of 
commodities means a rise of the exchange-value of money in 
commodities ; and the rise of people's earnings in money does 
not mean a fall of the exchange-value of money in labor, there 
being no such exchange-value, money not being exchanged for 
labor ; but it means a fall in the cost-value or esteem-value of 
money. Then what "value" is it of money that has remained 
stationary ? Surely there is no value that is a mean between 
exchange- value and cost-value (or esteem-value). 

Again, if the prices of commodities should for a time remain 
constant on an average, and if also the average money-earnings 
of workers remain stationary, the same result would be obtained 
from the mixed method of measuring the " value " of money — 
namely that the " value " of money has remained unchanged. 
Yet, in this case, both the exchange-value and the cost-value or 
esteem- value of money have remained unchanged. Thus the 
mixed method will give the same result in regard to the " value " 
of money, even though the sejiarate measurements of the ex- 
change-value and of the cost-value or esteem-value of money 
give different results. The separate methods, however, give us 
another result not indicated by the mixed method. In the last 
suppositional case it is evident that the cost-value and esteem- 
value of commodities in general have remained stationary. But 
in the first actual example it is plain, from the separate measure- 
ments, that the cost-value and esteem-value of commodities in 
general haA'c fallen even more than the cost-value and esteem- 
value of money. For a fall of the esteem-value of money with 
a rise of its exchange-value, means that men are obtaining 
money more easily and that their money is purchasing for 
them more commodities, that therefore commodities are being 

W. Holt, Interest and a27preciation, Hound Currency (Reform Club), New York, 
Nov. 15, 1898, p. 368. 

9 



130 SELECTION OF PARTICULAR EXCHANGE-VALUES 

obtained still more easily than money, and hence their cost-value 
and esteem- value have been falling still more rapidly than those 
values of money. The influence of falling prices and of fallmg 
esteem-value of money is cumulative upon the esteem-value of 
commodities — in our estimates of those things. But the mixed 
method, which indicates merely that an anomalous " value " of 
money is stationary as well in the one case as in the other, does 
not distinguish, does not inform us, whether connnodities are 
falling in cost-value, or esteem-value, or whether they are sta- 
tionary. And still another exactly opposite example is conceiv- 
able, in which money-earnings might fall and prices rise, in which 
therefore the cost- value and esteem- value of money are rising and 
the exchange- value of money is falling, where the mixed method 
of measuring might also indicate constancy in the " value " of 
money. Thus on three very different occasions the mixed method 
would give the same answer, although on two of these occasions 
the exchange-value and the cost- value (or esteem-value) of money 
are acting in diametrically opposite ways, and in the other they 
are standing still in the mean. Also this method might give 
the same answer no matter how much the exchange-value of 
money is falling, provided the esteem- value of money is rising 
sufficiently to counter-balance, or vice versd. The utter worth- 
lessness of such a method, which mixes up distinct things, is 
apparent. 

If such a mixed method is worthless when we carry out the 
one-half of it thoroughly, by including m the list all earnings, 
it is d fortiori worthless if we execute this half of it imperfectly 
by including in the lists only wages, and still more if only some 
wages. 

§ 8. There has really been gross confusion of thought in the 
recommendation of this method — an extension to a whole of 
what belongs only to a part. Considering only the case of per- 
sons who have fixed incomes and who spend some of it in buy- 
ing commodities and some of it in hiring servants, economists 
have seen that if the prices of commodities fall and the wages of 
servants rise, there is some tendency here toward compensation 
and balancing. In attempting to measure the purchasing power 



WAGES EXCLUDED lol 

of these incomes, therefore, it would be j)roper to take account 
of the wages of servants along witli the prices of goods. But 
then the prices of goods to be used, in order to observe parallel- 
ism, should be retail prices. And investigation should be made 
as to how much, on the average, is spent on goods and how 
nuich in wages, in order to weight goods and wages accordingly 
— probably with the result of finding the weight of wages to be 
relatively small, wherefore, for the counterbalancing, the rise of 
wages would have to be considerably greater than the fall of 
prices. In doing this, however, all that is accomplished is the 
construction of a special standard for a })art of the community. 
Then, forgetting the limitation of this procedure, and wanting to 
use wholesale prices, and to form a standard for the whole commu- 
nity of producers (including the servants) and consumers, and 
therefore to weight labor as equally important with the products 
of labor, some economists have assumed that a compensation 
which exists for a small part of the community exists for the 
community at large. The compensation exists, in full plenitude, 
only in the case of a few persons who are, so to speak, con- 
sumers both of commodities and of services, and are not them- 
selves producers or ser\4ce-renderers. In the case of the per- 
sons who render the services, or ^vho produce goods, Avithout 
themselves hiring other servants, instead of compensation, there 
is cumulation, for they earn more money and their money pur- 
chases more goods. While in the case of other producers and 
service-renderers who are also employers, the cumulative influ- 
ence generally exceeds the compensatory. Moreover, the wages 
of laborers hired for producing goods for sale, as before re- 
marked, are already included in the prices of the products. 
Now if these wages rise while these prices fall, either this change 
is at the expense of the employers, causing their profits to fall, 
and so not affecting earnings in general, or both the rise and the 
fall are compensated to the employers by cheapened methods of 
production and imj>rove(l machinery — that is, there is compen- 
sation of an entirely different sort. 

^ 8. Of course in makinp; the final mensurement of the esteem- 
value of commodities in general, we must count earnings as 



132 SELECTION OF PAETICULAE EXCHANGE- VALUES 

equally important with the prices of commodities. But this is 
because earnings wholly belong in the measurement of the cost- 
value or esteem-value of money, and prices wholly belong in 
the measurement of the exchange-value of money. Then, when 
a single result is obtained from the two measurements, equal 
importance is to be attached to each. 

But in the mixed method it is impossible to state what ought 
to be the weighting of commodities and of earnings as wholes 
relatively to each other. This problem has simply been ignored 
by most of the advocates of the mixed method. A couple of the 
early ones ^ weighted wages at about one third of commodity- 
prices, without telling us why, but apparently having in mind 
some estimate of the relative amomit of money paid m wages and 
in other expenses by the then dominant class of landlords, or by 
their farmers.'^ Recent writers have been less definite. Some 
have implied that even wages, let alone earnmgs, should be 
treated as equally important with the prices of all commodities ; 
while others have even asserted that greater importance should 
be assigned to wages than to prices. Of course these writers do 
not mean that people in general spend as much or more money 
in paymg servants than in purchasing goods. What they huve 
in mind is an indistinct notion that what they ^^'ant to measure 
is rather the cost-value or esteem-value of money. Then, of 
course, they should give greater weight to wages than to prices, 
for the simple reason that prices should not be counted in that 
measurement at all. Prices no more belong in the measurement 
of the cost-value (or esteem-value) of money than do wages in 
the measurement of the exchange- value of money.^ No wonder, 
then, that the persons who try to combine the two distinct 
measurements of the two distinct values are at a loss as to the 
amounts of importance they should attach to prices and to wages, 

^ Evelyn and Young. 

' Some time before, Cantillon said it was a common opinion in England that 
of a farmer's income a third goes to his landlord, a third to himself, and a third to 
his laborers, Essai sur le commerce, about 1732, Harvard ed., pp. 159-160. 

^ Among the writers above cited as mixing up these things, GiiFen at times in- 
clines to recommend treating of wages and salaries in a separate table, B. 55, p. 
131. But he does not recognize this as forming part in the measurement of some- 
thing else than exchange-value. 



wa(;es exci>uded 133 

and waver between them according a.s the idea of exchange-value 
or the idea of cost-value predominates in their minds. 

§ 10. Slightly different, though at the bottom similar, is the 
position held by a few economists and many politicians to-day 
that wages are a better measure, or criterion, of the " value " of 
money than the prices of commodities. Here the opinion seems 
to be that we may make one measurement of the " value " of 
money by means of prices and another by means of wages, with 
different results, and that of the two the measurement by wages 
is the better or more trustworthy. This position is uai've. The 
measurement of the value of money by wages alone is wrong, in 
the iirst place, because wages are not the only earnings. And, 
in the second place, a correctly performed measurement by earn- 
ings Avould be the measurement of the esteem-value of money ; 
wherefore, as such, it cannot be compared with the measurement 
by prices, which is not a measurement of the esteem-value of 
money at all. And the measurement l)y prices is the measure- 
ment of the exchange-value of money ; wherefore, as such, it 
cannot be compared with the measurement by earnings, which 
is not a measurement of the exchange-value of money at all. 
Each of these methods of measuring the " value " of money, if 
carried out in the best manner possible, is the best in its kind. 
And neither has a better claim than tlie other to be the better 
measurement of the " value " of money singly, since the one 
kind of value is as much value as tlie other.^ 

® It is hardly necessary to notice that sometimes the same economists who 
make the above comparison, and sometimes others, also bring in still another 
measurement of the " value of money " as equally good as, or even as better than, 
either of the preceding. This is the measurement of something which is not value 
at all in any of the* economic senses of this term, but to which this term is applied, 
in connection with a similai'ly uneconomic meaning of the term " money," in the 
slang of the "street." For certain classes of business men often mean by "money" 
loanable capital, and by the "value" of this "money" the rate of discount. 
Naturally the measurement of this "value of money " is by the rate of discount 
itself; and such a measurement is a perfectly good one of the " value of money " 
in these meanings of the terms. But it is not at all a measurement of the " value " 
of money in any of its economic senses; and so cannot properly be brought into 
comparison with any of the methods of measuring any of the economic values of 
money. — There is even another uneconomic meaning of "value," as used in the 
old monetary science, namely the meaning of "intrinsic value" in the case of 
metallic money, this being the fine metal in the coins. The measurement of this 
" value " is the simplest thing in the world, being made by finding the weight of 



134 .SELECTIOX OF PAIITICULAR EXCHANGE-VALUES 

§ 11. Of course if either of these methods is thrust forward 
into the province of the other, it should be driven back. But 
what happens mostly is that one person in speaking of " value " 
is prone to think only of the one kind of value, and then he 
claims that the measurement properly applicable to that kind of 
value is the better measurement of " value." He ought really 
to say that it is the only measurement of " value," and dismiss 
the other altogether, confining himself to the one meaning of the 
term he uses ; only this he cannot do, because the other mean- 
ing of " value " always does lurk in his mind. The best and 
only thing to do is to distinguish what kmd of value we are 
treating of, and to assign to each kind its own proper kind of 
measurement. Otherwise when two persons meet each other in 
the combat of opinions, who happen each to be thinking of a 
different kind of value although they are both talking simply of 
" value," they will be apt to fall into behavior very much like 
that of the tAvo knights on the opposite sides of the shield. 

And yet, to repeat, unlike those two knights, each of these 
contending parties has an inkling of the other's meaning, and 
adopts it himself at times. Here is the fundamental explanation 
why an}' economists have advised such a mixing of incompatible 
elements in the measurement of the " value " of money. Using 
simply the one term " value," they have not distmguished which 
kind of value they were seeking to measure, and so have allowed 
elements proper in the measurement of distinct kinds to creep 
into a single measurement, Avhich then is really a measurement 
of no one kind of value. Yet, as we have already had occasion 
to remark, many of the most prominent economists have af- 
firmed that when they used the term " value " they intended to 
refer only to " exchangeable value." And in truth most of the 
efforts of economists have hitherto been directed to measure 
principally the exchange-value of money, with a little infiltra- 
tion of cost-value elements. But cost-value itself, freed from 
any other kind of value, no one has ever yet attempted to meas- 
ure, or so much as mentioned as a desirable object of mensu- 

the coins aud allowing for the alloy. It would be no more absurd to bring this 
into the comparison and to say that the best measurement of the "value" of 
money is by the weight and fineness of the coins. 



WAGES EXCLUDED 135 

ration by it.selfV" Thus the mensuration of exchange-value has 
been brought well along, while the measurement of cost-value 
has hardly so much as been begun. There have been, and es- 
pecially nowadays are, a number of economists who look upon 
cost -value as a more important kind of value than exchange- 
value, — who, for instance, maintain that money, the recognized 
measure of " value," should be a fixed standard rather of cost- 
value than of exchange-value. Upon these, then, the duty is 
incumbent of instituting the attempt to measure the constancy 
or variation of the cost- value, or esteem- value, of money. 

The present work, to repeat, is concerned only with the men- 
suration of exchange- value. 

1" To be sure, Ricardo and Malthus and others, we have seen, were mostly in- 
terested in the measurement of cost-value or of esteem-value. But they always 
expounded their procedure as a search after the " real " measure of " value in ex- 
change," so that the measurement of exchange-value always interfered with their 
measurements of the other kinds of value. 



CHAPTER V. 

MATHEMATICAL FOEMULATION OF EXCHANGE- VALUE 
RELATIONS. 

I. 

§ 1 . For working out our problem of measuring the exchange- 
value of anythiug by combining its particular exchange-values 
into its one general exchange-value, or the many variations of 
these into one variation, the aid of mathematical formulation is 
indispensable. 

The money-unit, being our unit for measuring particular con- 
temporaneous exchange- values at the same place, has become also 
our practical small unit for measuring the comparative exchange- 
values of things in different places and their constancy or varia- 
tion at different times. But it is a variable unit, comparable 
with metallic or wooden foot-sticks, Avhich vary m length at dif- 
ferent temperatures and which also wear down, except that the 
money-unit may vary in exchange-value indefinitely in either 
direction, and so is a much less trustworthy practical unit. To 
be precise, it is not the money-unit itself, not the coin-unit — the 
material thing, — that is the unit of exchange-value, as also not 
the material stick is the unit of length ; but as the length of a 
particular stick is the unit of length, so the exchange-value of 
the money-unit is the unit of exchange- value. ^ Now just as the 
survevor corrects the variations of his measuring instruments in 
their attribute of length, so we want to be able to correct the 
variations of our economic measurmg instruments in their attri- 
bute of exchange- value, and wish to construct a true and unvary- 
ing unit underlying our practical units, the same in all places 

1 That, strictly speaking, we measure the value of things not by money, but 
by/value of money, see Prince-Smith, op. cit., p. 369; Knies, Das Geld, 2d ed., 
1885, p. 150; of. Rossi, op. cit., pp. 145, 157. 

136 



OF EXCHANGE-VALUE RELATIONS lo7 

and at all times. Some of the older writers on monetary mat- 
ters asserted that because money is the measure of other things, 
we cannot measure it.^ This is a false position in metrology, 
unhappily not yet wholly abandoned.^ All our practical meas- 
ures must themselves be measured — and not only our ordinary 
measures, but also the so-called standard measures.* In fact, 
metrology itself is the science of measuring our measures. Of 
course, the measurement of our measures is different from the 
measurement of other things. We measure other things by com- 
paring them with our measures — a simple operation. We meas- 
ure oiu' measures by comparing them with other things, prefer- 
ably ^vith some selected other things — a complex operation. 
The object of measuring other things is to find their relative 
sizes, as well as the constancy or variation of their sizes, by find- 
ing their sizes compared with one given and constant size, the 
measure used. The object of measuring our measures is to find, 
except in the case of different units of the same measure, only 
their equality or inequalities and their constancy or variations. 
This is the labor we have before us in regard to exchange-value. 
We want especially to measure our measure of exchange-value, 
money. 

As being the simpler form of our problem, we may devote 
most of the enquiry to the attempt to measure the constancy or 
variation of the money-unit in exchange-value at one place 

2 Javolinus in Digest, XLVI., I., 42 ; followed by Molinaeus and Biidelius (in 
Thesaurus's collection of monetary tracts, Turin 1(309, pp. 23(3,461). — The doc- 
trine was an inference from Aristotle's statement that money measures all things 
{Eth. N. V, V, 10). Hence Turgot also said that money can be measured only by 
other money, op. cit., p. 76. 

' L. A. Garnett, The crux of the money question; has gold risen f Forum, 
New York, Jan. 1895, p. 581. — And Mannequin has gone so far as to say not only 
that we do not measure our measures, but that it is not needed that measures be 
stable, this being a prejudice, and even, as measures should be like the measured, 
they ought to be variable ! op. cit., pp. 28, 70, Uniformite, monetaire, 1867, pp. 10- 
11 11., La monnaie et le double etalon, 1874, p. 39 (quoted by Walker, 3Ioney, p. 
281 n). 

■* That money is measured by other things was said by Montanari, Delia nioneta, 
1683, ed. Custodi, p. 91 ; Galiani, op. cit.. Vol. I., p. 52 ; Condillac, Le commerce 
et le gouvernement, 1776, ed., 1821, p. 93; (cf. Adam Smith, op. cit., p. 190) ; 
Ricardo, p. 293 ; Levasseur, B. 18, pp. 137-138. — McCulloch concluded that 
money is therefore not a measure of value, as it no more measures other things 
than it is measured by them. Treatises and essays, 1859, p. 10. 



138 MATHEMATICAL FOEMULATIOX 

through the course of time. The consideration of two periods 
will generally be sufficient. And for convenience we may sup- 
jjose, at the commencement, that all the classes of things are of 
equal exchange-value importance, so that we may employ even 
weighting, which is much the simplest to deal with first. 

§ 2. Let us represent the money-unit by the capital letter M, 
and the habitually used mass-units of various classes of things 
— say a bushel of wheat, a pound of sugar, a hundredweight of 
iron, a ton of coal — by the capital letters A, B, C, and so on. 
Or if the reader prefers, he may understand by these letters al- 
ways the same mass-unit — say, always a pound of wheat, a 
pound of sugar, a pound of iron, a pound of coal ; — for we shall 
see that in most cases no difference results from adopting either 
plan. From henceforth, the capital letters being appropriated 
for this purpose, we can no longer refer to the classes themselves 
indefinitely by the plain capitals ; but we may do so by putting 
the capitals in brackets, so that, for instance, we shall say that A 
represents a mass-unit of [A] , a class of things.'' AVe may sup- 
pose that at the first or basic period M will purchase a A, b B, 
c C, and so on — a, b, c, indicating the numbers (above, be- 
low, or at unity) of the mass-units of the classes which one M 
will purchase. Using the sign ==, as before, to express equiva- 
lence, we may express this state of things as follows, 

M = aA== 6B = eC=== 

At the second period we may suppose that M will purchase 

a'a A, b'b B, c'c C, and so on — a'a, b'b, c'e, representmg the 

new numbers of the mass-units of each class now purchasable 

with M ; wherefore a' , b' , c' , , according as they are at unity 

or as they depart from imity, above or below, express the con- 
stancy or variation, rise or fall, of the quantity of each class 
which one ]\I will purchase at the second period compared with 
the first, consequently the constancy or variation, rise or fall, of 
M's particular exchange-values in each class." It is plain that 

^This device of notation is borrowed, with a slight change, from Walras. 

^ E. g., if one dollar at first purchases 2 bushels of wheat (A, for instance, repre- 
senting 1 bu. of wheat) if it later purchases 2 bushels of wheat, then a = 2, a'a = 3, 
and «' = | = l2, which indicates a rise of 50 per cent, in M's exchange-value in 



OF EXC'HAX(n:-VALrK RELxVTIONS 189 

the fiig-ures «, b, c, will be various according to the mass- 
units which happen to be chosen, but, the mass-units once 
chosen being of course used at both periods, the variations repre- 
sented by a', h' , c' , are the same, no matter what be the 

mass-units used. Now at the second period we have 

M - «'« A : 6'6B = e'eC = 



These expressions, however, are not equations. They do not 
express equality between quantities of the same sort, but equiv- 
alence (/. e., equality in one kind of quantity, exchange-value) 
bet^\•eeu quantities of another or other kinds (weight, capacity, 
etc., of diiferent things). Therefore they are not satisfactory. 

§ .'>. Let us proceed ane^v by representing the general ex- 
change-value of M, of A, of B, of C, , by the italicized capital 

letters M, A, B, C, These exchange-values Ave shall need 

to distinguish at the two periods, Avhich we may do by ap- 
})ending a small number to them. When we Avish to distinguish 
between general exchange-value in all other things and general 
exchange-vakie in all things (including the thing itself), Ave may 
do so by appending a small o, or a small a. Thus llg^ means 
the exchange-value of the money-unit in all other things at the 
iirst period. Therefore the expression M === aA may (by Prop- 
osition VII.) be changed into il/j = aA^, AAdiich means either 
that the general exchange-A'alue of M is equal to the general 
exchange- value of a A, or that the general exchange-value of M 
is a times the general exchange-A'alue of A, at the first period. 
We maA' then haA'e 



31^ = ciA^ = bB^ = cC; = .... 
and 

31, = a' a A., = l/bB., = c'cC: 



Here Ave have equations proper ; for the terms all refer to equal 
homogeneous quantities, namely exchange-A'alues, although these 
are attributes possessed by different quantities of various articles 

[A]. If the figures were reversed, we should have a' =|, indicating a fall of .33J 
per cent, in M's exchange-value in [A]. If M purchases the same quantity at 

both periods, a' a = a, whence a' = - =1, indicating constancy. 



140 MATHEMATICAL FORMULATION 

variously measured as regards their masses. To say the ex- 
change-value of an ounce of gold is equal to the exchange-value 
of twenty bushels of wheat is to express equality as much as to 
say the weight of a bushel of wheat is equal to the weight of so 
or so many cubic inches of water. It is well known that a 
pound of feathers is as heavy as a pound of lead." 

These true mathematical equations admit of mathematical 
treatment. Thus from the equation 31^ = aA^ we may derive 

A, = — 31. ; and from 3L = a'ciA^, aA„ = — , 3L or A., = -y— i)/". 

1 a 1 2 2' 2 ^^/ 2 2 f^f^f^^ 2 

— which mean that a certain quantity of [A] , and consequently 
[A] in general, has varied in exchange- value ^ compared with 

[M] by — , the reciprocal of [M] 's variation in [A] (cf. Prop- 
ositions IX. and XIII.). If we knew that M2 = M^ we should 
know that a'ciA^ = aA^, whence A.-^ = — A^ which means that 
A, or [A] in general, has varied in general exchange- value by 
— (in agreement with Proposition XXXIII. ). Or if we knew 

that 31^ has varied from 31^ in a certain projsortion, we should 
know that a'aA^ has varied from aA^ in the same proportion, 
the general exchange- value of A, and of [A] , thus undergoing 
a double variation (in which the one may enhance, lessen, neu- 
tralize, or outdo the other). As yet, however, we do not know 
the relation of 31^ to Ji,, this being what we want to find. 

Still, for this purpose, the expressions are not serviceable in 
the meanings so far ascribed to them. For we cannot combine 
the general exchange-values of diiferent things in order to find 
therefrom the general exchange-value of any one thing. What 

' Bourguin says there are no equations between exchange-values, because ex- 
change-Talueis not intrinsic, B. 132, p. 38. The reason is not adequate. We have 
seen that even weight is not intrinsic. Certainly wealth is not ; yet two men may 
be equally rich. 

" If we are dealing with 31^, the general exchange-value in which [A] so varies 
is likewise A^. But if we are dealing with M^, the general exchange-value in 
which [A] so varies is not A^, but a general exchange-value of [A] in all things 
other to money, i. e., in all commodities, including itself. Another distinction 
of a similar nature will be noticed in 3 4. 



OF exchange-valuj: relations 141 

we can so combine are the particular exchange-values of the 
thing in question in the other classes of things. Now by Prop- 
osition VII. a thing's particular exchange-value in anything else 
is equal to its general exchange- value ; and by Proposition VI. 
the thing's general exchange-value is equal to the general ex- 
change-value of the quantity of the other thing it exchanges for, 
by which (juantity of that thing, according to Proposition I., its 
exchange-value in that kind of thing is measured. Therefore a 
thing's particular exchange-value in another thing is equal to the 
general exchange-value of that (piantity of the other thing it is 
measured by. Hence the expression aA^ may represent not 
only the general exchange-value of a A, but also the particular 
exchange-value M in [A], at the first period. And so the 
others. But it is necessary that the symbols should distinguish 
whether they refer to the general exchange-values of things, or 
to the particular exchange- value of M in those things. AVe may 
make them do so by always placing them in marks of paren- 
thesis, thus («^4), (bB), (gC), , when referring to the partic- 
ular exchange-value of M in [A], in [B], in [C], , at the 

first period. Then the expressions for the particular exchange- 
values of M in these things at the second period may be 

written either (a'aA), {b'bB), (c'cC), , or a'{aA), b''{bB), 

c'{cC), The first of these forms would be merely the ex- 
pressions for the particular exchange-values of M in [A] , in [B] , 
in [C] , , as measured by the quantities of these things it ex- 
changes for at the second period. But the second, while mean- 
ing this, mean, and express, something more, namely that the 
particular exchange-value of M in [A] , for instance, is a' 
times what it was before, that is, at the second jieriod a' times 
the particular exchange-value of M in [A] at the first period ; 
wherefore in this form of the expression obvit)Usly — and also in 
the other — the contained expression {a A) still refers to the old 
exchange-value of M in [A] ; and so with the rest. Hence it 
is no longer necessary here to number the expressions A, B, 
C, .' 

^ If we knew that Mq2 = ^J^oi^ then we should know that aA, if it refers to the 
general exchange-value of a A, refers to something different at the second period 
from what the general exchange-value of « A was at the first period, that is, it 



142 MATHEMATICAL FORMULATION 

We then have 

if, = (ciA) = (bB) = (cC) = , 

which may be more specifically written 

Moi = {a A) = (bB) = {cC) = to n terms, 

n representing the number of all the (equally important) classes 
of things employed. This means that the general exchange- 
value of the money-unit in all other things is equal to every one 
of its particular exchange-values, which are stated as they are 
at the first period. And we have 

Jfo, = {a'aA) = (b'bB) = (c'cC) = to n terms, 

which similarly expresses the equality between the general ex- 
change-value of the money-unit in all other things and every 
one of its particular exchange-values, which are stated as they 
are at the second period. But if we Avrite this as follows, 

Moo = a'(aA) = b'{bB) = c'{cC) = to n terms, 

it expresses the same equality, but states the particular exchange- 
values at the second period as they relate to what they were at 
the first period. 

§ 4. Using these running equations, we can no^v form expres- 
sions for the combination of the particular exchange-values of 

refers to the general exchange-value of a A at the second period, wherefore it should 
be distinguished by a number, viz., a^^o. But in referring merely to the particular 
exchange-value of M in [A] when we do not know what the relation between 
Mq,^ and jJ/qj is, but are seeking it, we are using the particular exchange-value of 
M in [A] at the first period as our standard, and must look upon the particular 
exchange-value of M in [A] as changed at the second period by «', and therefore 
we must take {aA) as the constant and a' {aA) as the variable. But {aA) alone 
is not our whole standard ; it is only part of the whole standard composed of all 
other things. Yet we have to treat each of these separately as a standard, before 
we can unite them. Then after we have united them, and when we use the whole 
composed of them as the true standard, it is no contradiction with the preceding 
that we may find aA at the second period to be different from aA at the first. The 
particular exchange- value of M in [A] has changed by a' (while the pai-ticular 

exchange-value of A in [M] has changed by —) ; yet, in the ease supposed, the 

particular exchange-value of M in [A], now a' (aA), is of the same magnitude as 
was the particular exchange-value of M in [A] when it was {aA), because at 
each period it is equal to the general exchange- value of M, which is supposed to 
be unchanged. The contradiction is only apparent, because of the change in the 
standard used. Cf. Chap. II., Sect. I., ? 1. 



OF EXCHANGP>YAIATE RELATIONS 143 

M in every other thing into its one general exchange-value in 
all other things. We can, in fact, mathematically form several 
such combinations, but may content ourselves with three. To 
begin with the first period. From 

J/^j = {a A) = (bB) = (eC) = to n terms 

we perceive that the sum total of the exchange-values of nM 
will be 

nJI,)^ = {a A) + (bB) -\- (eC) -\- to n terms ; 

whence 

J/^j = 1 |((U) -j-{bA)-^ (eC) + to n terms }.^" (1, i) 

Again, the serial equation may be converted into this, 

1111 , , 

= = = = to 71 terms, 

Mo, (aA) (bB) (cC) 

(which means that the exchange- value of a A in [M] is equal 
to the exchange-value of b B in [M] , and so on, and to that of 
M in [M], in agreement with Proposition IV.). By the same 
treatment this yields 

Hill ^ , 

= J 1 L to n terms, 

Moi {(lA)^ (bBy (cC)^ 

whence 

Mfn = Y~[ J J \ r • (^' -) 



w ( \ + 77~m + 7~r^^ + ton terms 

71 [ {(f'A) [bB) (eC ) 

Lastly, from the first serial equation, and also from its inverted 
form, the ttjtal product of the exchange-values of M taken n 
times will be 

J/j; = (a A) ■ {bB) ■ {cC) to ?i terms, 

whence 

^%ji = {/{aA) ■ {bB) ■ (cC) to n terms. (1,3) 

1° With different notation, and also with reference to the quantities of tilings 
instead of their exchange- values, this form of the formula was, perhaps first, given 
by Prince-Smith, op. cit., p. 372. Prince-Smith reached this formula in some- 
what the same way as it is here reached ; but he failed to see that there are other 
formulie equally well deducible from the same original running formula. The 
idea expressed by this formula is also alone advanced by Nicholson, B. 94, pp. 
307-308, and by Fonda, B. 127, p. 161. 



144 MATHEMATICAL FORMULATION 

Naturally the serial equation for the second period may be 
treated in the same way, and, likewise with even weighting, in 
its first form, it will yield, with omission of the superfluous " to 
n terms," 

Mo, =^\{a'aA) + {h'hB) + (c'cC) + I, (2, i) 

^-^--= if 1 1 -^ T- T ' (^' ^) 

71 I {a'aA)'^ {b'bB)^ {c'cC)^ j 

Mo, = V(«/aJ.)-(6'6i?)-(c'c(7) • ■". (2, 3) 

What is here done for M, be it here parenthetically remarked, 
can be done for any of the other classes. For example at the 
first period, if Ave want to express the general exchange-value 
of A directly in all the others, we may begin by representing 
the general exchange- value of « A by a A 01, and then {M) would 

represent the exchange- value of a A in money, and (bE), (cC), 

will continue to express the exchange value of a A in [B] , in 
[C], Then from 

a A or = {M) = (bB) = {cC)= to w terms, 

we can derive 

aAox = -{{M) + {bB) + (cC) + to n terms}, 

as in the first Avay ; and similarly in the other tAVO Avays, Avhich 
may be omitted. This formula Ave could have derived directly 
from formula 1, by simply substituting a A 01 for Mqi and [M) 
for {aA) ) and so on in the other omitted Avays. And from this 
formula Ave deriA^e 

^o^ = -^^{{M) + {bB) + {eC) + }. 

Similarly for the second period avc can get 

a'aAo, = ^{W + (b'bB) + (c'cC) -\- }, 

and 



OF EXCHAN(;E-VAI.rE EEI.ATIOXS 145 

It floserves, however, to be noticed that if A\e should use these 
formulae to calculate the variation of the general exchange-value 
of A in the way to be described presently, we should obtain a 
result diiferent from that obtained by calculating first the vari- 
ation of the general exchange-value of M and then the variation 
of A by means of its variation in the varied money. For these 
two methods use diiferent standards, the class A itself forming 
part of the standard in the latter method, but not in the former, 
and money forming part of the standard in the former, but not 
in the latter. (Of. Propositicnis XII. and XIII.) 

The interpretation of the above three kinds of formula? is as 
follows. They each express primarily a way in Avhieh the ex- 
chanc/e-value in all other things of the money-unit (or of the mass- 
unit of anything else) is to be conceived, the first, that it is to be con- 
ceived as equal to the sum of its exchange-values in every one of the 
other things, or to the sum of all its particular exchange-values, di- 
vided bif their number ; the second, as equal to the i-eciprocal of the 
sum of the reciprocal of all its particular exchange-values divided by 
their number ; the third, as equal to the n"" root of the product of 
all its particidar exchange-values, n in number. 

Each of these formulae is nothing but the expression of a 
mathematical mean or average between two or more equally im- 
portant quantities — the first, of the arithmetic, the second, of the 
harmonic, the third, of the geometric. And each is perfectly 
true as an expression of its own kind of average drawn between 
the quantities given. But not more than one of these averages 
— and perhaps none of them — can be the proper or correct aver- 
age to draw for the purpose avc are putting it to. Having three 
ways of combining or averaging a thing's particular exchange- 
values into its one general exchange-value in all other things, 
we have an embarras de richesses, and much of our subsequent 
labor will be to decide between these different forms. 

§ 5. Now our purpose being to compare the two combinations 
or averages at the two periods, we may make this comparison by 
dividing the formula for the second period by that for the first. 
Here we evidently want that form of the expression for the 
second period which brings its contained particular exchange- 
10 



146 MATHEMATICAL FORMULATIOX 

values into relation with the particular exchange-values of the 
first period. The comparison then will be as follows : 

^ l{a\aA) + b'{hB) + o'{cC) + } 



02 

M ~ ^ ' 

^{(a^) + (65) + (cC) + } 

1 

IT i T 1 ^ 



(3, 



1( 1 J_ J_ ] 

n^{aA)^ {hB)^ {gC)'^ i 

if 02 v''^il{ciAyh'{bB)-c'{eC) .g 

^ ~ O'iciA) ■ (bB) ■ (ccy^- ■^ 

Here we have the formulse desired for the variation of the 
general exchange-value of M, but in forms more complex than 
necessary. At the first period the ex change- values (a^), (bB), 
(gC), are each equal to the exchange-value of one money- 
unit. The exchange-value of the money-unit we take for the 

unit of exchange- value. Therefore (avl) = (65)= (cC)= 

= 1 . Substituting this value of the terms m the denominators 
in each of the above equations, they all reduce to 1, thus : 

^^01=^(1 + 1 + 1 + ) = ^^(nl)=l, 

3f 1 1 1_1 

^'^01- 1 /I 1 1 \- i/,,\ - 1-^' 

4l+T+l + ) n[l) . 

Ji"„, = f/lll- = ^n/V" = 1. 



-01 



Therefore, all these denominators may be dropped. In the 

numerators the terms, if written (a'aA), (b'bB), (c'cC), , 

are also each equal to the exchange-value of the money-unit at 
the second period, and so the numerators also reduce to 1 — only 
this is a different unit. But, of course, we want the same unit 
to be employed in all the terms throughout the equations, and 



OF EXCHAxfjE-YALUE REEATIOXS 147 

the unit wanted is the unit employed in the denominators — the 
exchange-vahie of the money-unit at the first or basic period. 

NoAv in the numerators the terms (aA), (bB), (cC), , though 

no longer equal to the money-unit at the second period, still are 
expressions for the particular exchange-values of M at the first 
period, and so are still equal to the old unit. Therefore, as we 
employ this unit throughout, these terms, each = 1, may be 
dropped from the numerators also, and the equations, thus freed 
from useless symbols, reduce to the following : 

mA^'' + ''' + '■' + )' (^''^ 



-01 

31,, 1 



3ln 1/1 1 1 



a^) 



Here we have the workable forms of the formulae desired. 
In these forms the formulae each express one of the mathematical 
averages of the variations, from the first to the second period, of 

the exchange-values of M in [A], in [B], in [C], , that is, 

of the variations of all the particular exchange-values of the 
money-unit in every other class of things. This appears to be 
as it should be ; for a thing's variation in general exchange- value 
in all other things would seem to be made up of the variations 
of all its particular exchange- values, and therefore to be an 
average of some sort between all the particular variations. But 
we reached this position only througli really comparing two dis- 
tinct averages, in each of ^\'hich the hypothesis was that the 
classes are equally important — /. e., equally important at each of 
the periods. And in both these averages we used, in obedience 
to the principle stated in Chapter IV., Sect. V., § 9, the same 
number of classes — the same n. If we had used a different 
niunber of classes in each averaging, we could not have made 
this reduction to a common averaging. A¥hether we could make 
this reduction if the classes were unequally important at each 
period singly, but were equally important over both the periods 



148 MATHEMATICAL FOEMULATION 

together, is questionable. Perhaps it may be proper, perhaps 
not. By most writers on the subject, however, the question has 
hardly been raised. The fine distinctions as to the periods when 
the weighting is to be measured have not been noticed. Gen- 
erally the assumption has been made that we can average the 
variations of the particular exchange-values using a single weight- 
ing, even though the weights of the two periods separately are 
different. We may for the present ignore this question, and as- 
sume that we are justified in using— or are dealing Avith cases 
which permit us to use— even weighting in an averaging of the 
variations of the particular exchange-values. 

§ 6. The averages of the variations may be represented in 
still another way. Instead of representing the numbers of mass- 
units of the various classes purchasable by if at the second 

period by a'a, h'b, c'g, , let us represent them by a^, b^, e^, 

in distinction from which we shall represent the numbers 

purchasable at the first period by a^, b^, c\, • In this nota- 
tion a^, 62, C2, , taking the place of a'a, b'b, c'c, , are 

equal to them respectively ; wherefore, as a.^ = a a^, ^ = <^^ j 

and similarly ^^ = 6', -' = e', and so on. In other words, the 

^2 "2 ^2 
variations are represented, in the new notation, by ^, ^, -, • 

Therefore these new terms may take the places of a', b' , c' , 

in the formula 4, 1-3, and we have equally good and valid 
formulse in the following : 



Mar l/«i^i + 5 + 
■ n\a^ 62 c. 



(5, ^) 



if ,, - 1/ a, 6, c, • • 

§ 7. Any of these averages bemg adopted, if the result in any 
comparison of two periods is larger than unity in a certain pro- 



OF EXCHANGE- VALUE DELATIONS 149 

porti(in, this moans that the exchange- vahie of money in all 
other things is indicated by this kind of averaging to have risen 
in that pr()])ortion {c. ;/., if the result is 1.15, the rise is by 15 per 
cent.) ; if the result is less than unity in a certain proportion, 
the exchange-value of money is indicated to have fallen in the 
same proportion (e. g., if the result is .85, the fall is by 15 per 
cent.) ; and if the result is unity, the exchange-value is indicated 
not to have varied, /. e., to have remained constant. 

If the variations of all the particular exchange-values of jNI are 

the same, so that «'= 6'= c'= = ror -? = -?= -2=: _,. 

«1 ^1 ''l 

SO that this one quantity can stand for them all, the results of 
all the averages are the same, and are r, as shown by the fol- 
loAving : 

^^A^=l-(r +/• + .+ )=\nr) = r, 

Mq-^ n n 

Mo, 1 _ 1 _i_^. 



M,. 1/111 \ l/n\ 1 

-- + - + -+ -• 
n\r r r 



W) 



Mo, 



Mo, V -V ' -'- 

Thus all these results agree with the knowledge we already pos- 
sess that if a thing varies in all its particular exchange- values 
in a certain proportion, it varies in exchange-value in all other 
things in the same proportion (see Proposition XVII.). Un- 
fortunately, therefore, that piece of knowledge does not help us 
to decide between these methods of averaging. Of course, if 
there is no variation in any of the particular exchange-values, 

so that a' = b' = c' = = 1, or '!l = h^^^ = 1, the 

result in all cases is 1, meaning that there is no variation in 
the general exchange-value (in accordance with Proposition 
XXVII.). 

But if there is variation in at least one particular exchange- 
value, or a variation in at least one particular exchange-value 
divergent from all the other variations, — if there is the slightest 



150 MATHEMATICAL FORMULATION 

irres'ularity, — the results in the three kinds of averagmg always 
differ from each other. Between the three mathematical aver- 
ages the mathematical relations are well known. If the varia- 
tions happen to be such that the geometric average indicates 
constancy, the arithmetic average will always indicate a rise, the 
harmonic a fall. If the geometric average indicates a rise, the 
arithmetic will indicate a greater rise, the harmonic a lesser rise 
(perhaps constancy, perhaps a fall). If the geometric average 
indicates a fall, the arithmetic will indicate a lesser fall (perhaps 
constancy, perhaps a rise), the harmonic a greater fall. The 
geometric average is always between the arithmetic above and 
the harmonic below. In fact, when there are only two equally 
important terms used, the geometric mean is the geometric mean 
between the arithmetic and the harmonic means ; and in other 
cases the geometric average is near that mean.^^ 

It is plain that in none of the methods can constancy be in- 
dicated if there is only a variation of one particular exchange- 
value of money, or if all the variations are in the same direc- 
tion ; but only in the case of opposite and compensatory 
variations. This also agrees with what we already know (see 
Propositions XX. and XXVIII. ). Given a single variation, 
however, each of the methods, in order to indicate constancy, re- 
quires a different compensatory variation — the arithmetic the 
greatest if the compensatory variation be a fall and the smallest 
if a rise, the harmonic the smallest if a fall and the greatest if a 
rise, the geometric always an intermediate. 

II. 

§ 1 . It is easy merely to introduce uneven weighting into the 
formulse. We only have to remember what uneven weighting 
means. Uneven weighting means that one class is larger than 
another, that is, contains more individuals ; while classes evenly 
weighted are supposed to contain an equal number of individ- 
uals. What these individuals are we need not here again dis- 
cuss. Xow if a class contains twice as many such individuals 
as another, it virtually consists of twice as many classes as the 

11 See Appendix A VI. g§ 1-5. 



OF EXCHANGE- VALUE RELATIONS 151 

other ; and if we treat the other as one class we may treat this 
as two chisses. The double number of mdividuals which are 
combined mto the first class are so combined into one instead of 
into t^\'o classes only because they all have the same name and 
the same exchange-value. They are one class nominally ; but 
logically they arc t^vo homonymous classes — compared with the 
other as one class. Thus suppose the class [B] is twice, and the 
class [C] thrice, as large as the class [A] ; the state of things then 
is the same as if we had two classes both called [B] , in both of 
which M has always the same exchange-value, and three classes 
all called [C] , in all of which M has always the same exchange- 
value. If this be supposed to be the state of things at the first 
period, then at the first period the formulae would be as follows : 

^^m=^, \ {aA) + [bB) + {bB) +{cC) + (cC) + (cC) + to 71^'' terms | 

= ^^ I {ciA) + 2{bB) -f 3(cC) + to n terms|, (6, i) 

1 

-^^^'^ 1 f 1 , 1 1,1,1,1, „ ] 

r7^ laZ + 6^+65 ■^^' + c"C' + cC + ton- terms | 

1 

"1,2,3, - - r ' (6, 2) 

c71 + 6^ + cC+ tonterms | 

Mm -= "i^ {aA) ■ {bB) ■ {bB) • {cC) ■ {cC) ■ (cC) to n- term7 

= ''{/{aA)- {bBy^- {cC)^ to 71 terms; (6,3) 

in which n" represents the number of the virtual classes, or n 
enlarged to cover all the coefficients or exponents. 

If, then, at the second period the same relative conditions per- 
sisted, in spite of variations in the particular exchange-values, 
we should have formulie similar to these, with addition only oi 
the symbols indicative of the variations. Then the comparison 
is easy. We may, however, represent the weighting more 
generally by employing algebraic symbols throughout. Let a 
represent the weight of [A] , b the weight of [B] , C the weight 
of [C], and so on, all these weights bemg supposed to be the 

same at both periods ; and let w" = a -|- b -|- C -|- + to n 

terms, the same classes of course being used in each of the 
averages compared. Then w'e have 



152 . MATHEMATICAL FORMULATIOjST 

^ —,{ai.a'{aA)+'bb'{bB) + Cc'{cC)-j- to ?i terms} 

^^ — ,{a(a^)+b(65) + c(cC) + to w terms} 

1 
1 r a b c 

Mo, ~ 1 



(7,0 



, (7,0 



a b c 

+ 77"^ +;^rF^ + to 71 terms 



n"((a'^)^(65)^(c(7) 

^02_t/'^{«'(a^)}'- {b'{bB)Y-{G'{eC)Y to^iterms 

^01 \/{aAf ~{bBf ■ {cCy to n terms 

In all these, again, the denominators reduce to unity ; and 

also the expressions {a A), (bB), (cC), in the numerators 

are the same units. Therefore we have 

M , 1 

^ = ^, (a a' + b 6' + C c' + to n terms), (8, i) 

Mo, 1 



Mo, 



~" 1 /a b c \' 

^1^ + 6^ + ? + tontermsj (8,.) 

Mo, ^ j^/^,a . ^/b . ^/c . ^^ ^terms. (8, s) 

Or if we prefer the other system of notation to represent the 
variations, we have 

Mo, 1 / rto . , b^ c. 



Mo, 



= ^{^%-^''\-^'l^ tonterms), (9;.) 



if., 1 



^-, ( a -^ + b 7-^ + C - + to n terms ) 

n" \ a^ 62 e^ / 

£=^©"(a"©° -^^- ^'"^ 

Here again are formulse representing averages of the variations 
of the particular exchange- values. The reductions by which they 



OF EXCIIAXGE-VALUF: ItELATIOXS 153 

are obtained suppose that the weights are the same at each period 
separately. But generally in practice the weights are diiferent 
at each period separately. Still if we are to average the varia- 
tions, we nuist employ only a single system of weighting — for 
we cannot, for instance, have a variation of two classes of [A] 
and a variation of three classes of [A] . We must have either 
a variation of two classes of [A] , or a variation of three classes 
of [A], or a variation of a number of classes of [A] that is a 
mean between the numbers of classes of [A] at each of the 
periods separately, as examined in the preceding Chapter. For 
the present we may assume that this course is a proper one, and 
that one of these weightings is the proper one. We mav, then, 
continue to represent the weight of [A] by a, and that of [B] 
by b, and so on, and their sum I)}' n", in Avhatever way these 
weights be calculated. 

The averages agree with much that we have already discov- 
ered concerning variations of exchange-value, and disagree ^vith 
nothing we yet know. Thus it is plain that in any of these 
formulse a change of av eighting without any change of exchange- 
values causes no change in the result ; for then the terms ci' , b' , 

Cl o c 
'\ -r , —, , iii'c all 1, and its does not matter how 



or 






large or small a, b, C, be, the results are always 1, indicating 

constancy in the general exchange-value of money. This is in 
agreement with Propositions XXVII. and XLIV. Also if the 
variations are all alike, so that a' = b' = c' = = r, or 

— =-7^'= — = = /•, no matter what be the weights, the re- 

suits are always r,^ in agreement with Propositions XVII. and 
XLV. 

In all these formulae the results are the same whether we use 

the weighting a, b, C, , with ?i" = a-|-b-|-C-f to n 

terms, or whether we use ra, /'b, /■ C, , with n" = /-a + r b 

-\- r C to n terms.^ This means that it is indifferent how 

large or small be tlie weights, provided they always maintain 

1 Cf. Appendix A I. i^?5, (5, II. jJ t3, III. ?G, V. ?."). 
2Cf. Appendix A I. j^ 11. 



154 MATHEMATICAL FOEMULATIOX 

the same relative sizes. Therefore the weights may be any 
numbers, larger or smaller than unity, mtegral or fractional. 
It is, then, not necessary that any of the classes should be taken 
as a miit class like [A] in tlie above numerical example, al- 
though our subsequent calculations might be slightly simpler if 
this were clone and the other weights reduced accordingly. In 
this case the coefficient of the unit class, and of all the classes 
equal to it, namely 1, must be counted in the composition of the 
term n". 

§ 2. It may, however, turn out that this course of averaging 
the variations on a single weighting is not right, no weighting 
bemg discoverable that will m all cases yield a demonstrably cor- 
rect result. If this be so, we shall not be able to make the re- 
duction of the comparison of the two averages, one at each period, 
to a single average of the variations, but shall have to retain the 
comparison of the two averages. This constitutes what we have 
called the method of double weighting. In our formulae we shall 
have to distingush a into a^ and a,, b into b^ and bg, and so on, 
representing the weights at the first and second periods respec- 
tively ; and consequently also n" into n^' and n^', although n 
must remain undivided. Then the formulae will become the fol- 
lowing : 

,^ —„\di,.ia' {aA) + b,6^(6-B) -{- C2C^(cC) + to n terms \ 

^'=--'-S: '- > (10. i) 

—j,\2i,i{aA) -|-bi(65) + Ci(cC) + to n. terms} 

1 

1 f a2 . b,_ , C2 , I 

J/Q2 _ n/' \a^[aA) ^ y(bB ) ^ c^jc C ) + to n terms j- 

mi- 1 ^ ' '' 

'k'{{^) + WB-) + W] + to^terms| 

Mo--> 'yU'{a^)l^'- ly{bB)]*'^- \c'icC)\^' to « terms 

y/ (a^)^' • {bB)*"^ ■ (cC)<=» • to n terms 

But here the denominators again all reduce to unity, and may 
be omitted ; and if in the numerators the terms (aA), (bB), 
(cC), be conceived as referring to the exchange-values of 



OF EXCHAXGE- VALUE RELATIONS 155 

the money-unit at the first period, and therefore, as mere units, 
be omitted, the formula? reduce to expressions f )r the averages 
of the variations with the weighting- of the second period — which 
is absurd. And if, then, we conceive of the terms a'(aA), b'{hB), 

c'(cC), as referring to the exchange- vahies in the money of 

the second period, whereupon they each become equal to unity, 
the luunerators also reduce to unity, and all these formuhe are 
useless. 

What we need is this. The terms a, b, c, , and a'a, h'h, 

c'c, , or better, a^, b^, c,, and a.^, b^, c^, , express the 

numbers of times the money-unit is more valuable than certain 

mass-units of [A], [B], [C], at the first and at the second 

period respectively. Therefore we want to average these ratios. 
But to average them properly we need to know how often they 
are repeated at each period. Let x^ and x^ represent the num- 
bers of times the mass-unit of [A] is repeated at each period, 
that is, the numbers of these mass-units that have been produced 
or consumed within these periods ; and let y^ and y.^, z^ and z^, 

represent the numbers of the mass-units of [B], [C], 

And let n/ = x^ -\- i/^ ^ z^ -j- to n terms, and n./ = x.^ -f y^ 

-|- ^2 + to n terms. Then the desired formulae are 

1 
If ^f{^2^2 + 2/2^2 + ^2^0 + to n terms) 

df±0-2 "'2 






,(.«V'i + //i^i + ^'i'"i + to n terms) 



_ ^t/(-¥^2 + yh + '^2^2 + to 71 term s) ^ .^^^ ^^ 

n^{A\a^ + 2/1^1 + ^1^1 + to n terms) ' 



1 



1 /.r., y„ z„ \ 

.r "(- + r + ~ + *'^ " *^'™^0 

3I0.2 n./\«, 62 c., J 

( il + ^ -f ^' -f to n terms 

(X 1/ z \ 

-1 -\-'^ M^ -\- to n terms ) 
JLA_^^j I , (11, .) 

/x If z \ 

n, 



h(- + r + - + to 71 terms^ 

V^2 ^2 ^*2 / 



156 MATHEMATICAL FORMULATION 



^^^0^ y/ a^- ■ hy^ ■ c{- to n terms 

These may also be written more fully by expanding the yi'q. 
The first so treated becomes 

^02 x^a^ + 2/2^2 + ^2^2 + •'^1 + 2/1 + ^1 + 



i¥oi x^a^ + y^\ + z^G^ + x^ + y^+z^+ 

which is the general formula for the method of double weighting 
with the arithmetic average. 

But here an important remark must be made. It is evident that 
in all these last formulae the results will be variously affected by 
the sizes of the mass-units we happen to choose. For if we choose 
large mass-units, the numbers of times they occur at each period 
(the expressions x, y,z, ) and the numbers of times the money- 
unit is more valuable than they (the expressions a,b,c, ) are 

smaller than the numbers of times smaller mass-units would 
occur and than the numbers of times the money-unit would be 
more valuable than smaller mass-units. In some of the terms, 
as in the parts within brackets in the second formula, these two 
changes neutralize each other ; but not in others, and never in 
the terms w/ and n^' , except only if all the mass-units be altered 
in the same proportion. But if the mass-units be variously and 
indiscriminately chosen, the terms n/ and n.J , not to mention 
others in some of the formulae, will variously relate to each other, 
and the results will be various, in a haphazard manner. Hence, 
it is necessary that some method of selecting the mass-units be 
established, as without such a method also these formulae are 
worthless. Such a method need not be stated in the formula, 
and then the above formulae become applicable only on the sup- 
position that the method of selecting the mass-units has already 
been found and applied. Or again it may sometimes be possible 
to introduce into the formula itself the method of selecting the 
mass-units. This has actually been done in some methods deal- 
ing with the mverted cases of price measurements. These we 
shall examine later. 



POSSIBLE EEROES 157 

III. 

§ 1. Certain ^variiings are needed against errors into which 
the use of mathematical formulae in this subject may lead us, 
unless we be careful. 

Two general i)rinci})les are obtained from the preceding in- 
vestigations. The one is that in cases when the weighting is 
the same at both the periods compared, it is immaterial whether 
we average all the particular exchange- values at the two periods 
separately and then find the variation of these averages, or 
whether we average all the variations of the particular exchange- 
values. The other is that >vhen the weighting is different, and 
different weighting is used in the averaging, at each perit)d, we 
can only find the variation of these averages, there being no one 
averaging of the particular variations that can use double 
weighting. We have seen, however, that we can dra^^' an aver- 
age between the two weightings ; and now we might use this 
single weighting in averaging the particular variations. But 
we shall find that this last operation will yield tlie same result 
as the comparison of the two separate averages only in peculiar 
circumstances. We therefore have ' before us two distinct 
methods. The one is to compare the averages separately drawn 
at each period. The other is to average the particular varia- 
tions. These two coincide only when the weighting actually is 
the same at both periods, or under other peculiar conditions. 
Even in the first case, when the weighting (reckoned according 
to the relative total exchange-values of the classes) is the same 
at both periods, there might be made averages at each period on 
other systems of Aveighting (^. g., according to mass-quantities), 
which might be different at each period, or if the same at both 
periods, might be so only in case the weighting, properly reck- 
oned, is different at each period. 

Thus the subject is very complex. Running through it all are 
the similarity and difference between comparisons of averages that 
have varied and averages of the particular variations — a purely 
mathematical question, which has never been fully studied even 
by mathematicians, let alone economists. Hence it has been nee- 



158 MATHEMATICAL FORMULATIOJST 

essaiy in this work to make such a mathematical study, to serve 
as a foundation for the various applications of the mathematical 
principles which we shall have to make in dealing with the 
averages of exchange- values and of prices, and with their varia- 
tions. This study is added as Appendix A. 

There are many difficulties engendered by the many complexi- 
ties in our subject, which we shall meet from time to time. But 
there are some simple errors into which we might easily slip at 
the commencement, and into which many persons have slipped. 
Some of them are even suggested by the second method of no- 
tation above given — a method which, in its general feature of 
distinguishing the exchange-values (or again the prices) of the 
same classes at the different periods only by different subscript 
numbers appended to the same letters, is the simplest and the 
most commonly used. Against the errors so suggested it is well 
here to give warning. These occur even when the weighting 
is the same at both periods — or when we are attempting to use a 
single weighting in averaging variations. Confinement may be 
made for the present to such cases. 

§ 2. A general error is this. Because in these cases the 
method of averaging the variations properly agrees with the 
method of comparing the averages, and because the former is the 
simpler method, we might be tempted to say of exchange- values 
what Jevons said of prices,^ that there is no such thing as an 
average of them, but only an average of their variations. In- 
deed, as all the particular exchange-values of anything at any 
time are always equal, not only the average of the particular 
exchange-values of money properly weighted at the first period 
is always unity, or IMq^, but at the second period also the aver- 
age of the particular exchange- values of money properly weighted 
will also be unity m the new unit, Mq^. Still there may be a 
different average of the exchange-values at ihe second period in 
the unit of the first period ; which is just what we want, both 
because we ought to conduct the whole operation on the same 
unit, and because it is convenient that the average at the first 
period, with which that at the second is compared, should always 

1 B. 22, p. 23. 



POSSIBLE ERRORS 159 

be 1 , as it forms the denominator in the expression for the com- 
parison. The process then is the following : At the first period 
we construct, as it were, a level of the particular exchange- values 
of money. This level breaks U]) at the second period into an 
undulating line, of the various heights of wdiich an average may 
be drawn and compared with the uniform height of the first 
level. This operation evidently gives both an averaging of the 
variations and a variation of the average at the second period 
from the level or average at the first. It is difficult, in fact, to 
conceive of an average of variations >vithout admitting an aver- 
age of the starting and of the finishing points. But the averages 
do not exist in nature, nor are the positions given upon which 
averages merely need to be drawn. The whole operation has to 
be constructed, and error will ensue unless it be constructed in 
the right way. Our guiding consideration must be that the 
variation of the averages must, in these cases, give the same re- 
sult as the proper average (of the same mathematical species) of 
the variations. This agreement is always obtained if we con- 
struct the average at the first period upon equal terms, so that 
the average itself is equal to them, and then use the same weight- 
ing in both methods. 

But the agreement is also obtamable by comparing separate 
averages at each period without the average at the first period 
being so constructed upon equal terms. This occurs always in 
the geometric average if we use the same weighting in both 
methods ; but in the arithmetic and harmonic averages never if 
we use the same weighting,^ and only if we use different weight- 
ing in each method, a definite relation existing between those 
which have to be employed to get the same result. For in the 
arithmetic and. harmonic averaging the inequality of the terms 
themselves at one of the periods forms part of the weighting in 
the comparison of th« separately made averages, but has no in- 
fluence on the weighting in the averaging of the variations ; 
wherefore, unless it is allowed for in the former, an apparent 
similarity in the weighting will really be a diflFerence, and real 

2 Except when all the variations are exactly alike, or when there are no varia- 
tions. See Appendix A I. § 6. As these cases rarely happen, and as we have no 
control in regard to them, they are not of much importance. 



160 MATHEMATICAL FORMULATION^ 

similarity can be obtained only by employing different apparent 
weighting. 

Now if we had at the start employed the form of notation later 

introduced, namely that of using a^, h^, c^, for the numbers 

of the mass-units of the classes purchasable with the money -unit 
at the second period, we might have fallen insensibly into the 
treatment of comparisons of averages employing even, or what 
we thought to be the proper uneven, weightmg, and to have 
supposed that we were then employing even, or the same proper 
uneven, weighting in averaging the variatioi]5; or if we happened 
to notice the divergence in the results, we might have been non- 
plussed, and perhaps have given up the subject in despair, as 
some economists have done. 

The errors we could thus fall into are different in the differ- 
ent kinds of averagmg ; wherefore these need to be treated sepa- 
rately. 

§ 3, Arithmetic averaging. — Intending to employ even weight- 
ing at both periods and in the averaging of the variations — or to 
disregard weighting, using " unweighted " averages — we might 
represent the average at the first period thus, 

M,, = l{{a,A)^{h,B) + {e,C)^ }, 

and that at the second period thus, 

^h. = 7J(M) + (^2^) + (^2^) + }; 

wherefore the comparison of these would be 
1 



Mo,- 

n 
which reduces to 



{(a^A) + {h,B) + {c,C) + ...,...} 
l{{a,A)-^{\B) + {c,C) + } 



M,, {a^A) -f- {b^B) + {c,C) -h 

Mo, - {a,A) + {b,B) -F {c^C) + ' 

the variation of the arithmetic averages being the same as the 
variation of the sums. Both of these comparisons of the averages 



POSSIBLE ERRORS 161 

^v'ould be perfectly correct, though meaningless without flirther 
treatment. Hereujion ^ve might have argued that, as A, B, C, 

seem to represent unity, or merely to make reference to the 

classes [A], [B], [C], , therefore they can be omitted, and 

so we should liave 

M^o _a.^^b., + c.,-\- 

M,,, " a^ + b^ + c, + -....• 

This argumentation, however, would be wrong, because it is 
not A, B, C, which are equal to unity (the exchange- 
value of the money-unit), but (nA), (pB), (gC), ; and 

A, B, C, do not refer to the classes, but, taken thus 

alone, to the general exchange-values of the mass-units of the 
classes. Then, again, if we dropped only the denominator as 
reducing to unity, the remaining form 

^;=^{(M) + (M) + (^.o) + } 

would be meaningless, because in the unit at the second period 
it too reduces to unity, and it contains no indication of the value 

of the terms (a.^A), (b.^B), {c.^C), in the original unit Mq^. 

We do, however, know that a^A = b^B = c^C= = Ij 

wherefore A = , B = ,C= , and so on. Substituting 
these, we get 

Moo l/((, ^2 ^2 



Mo, 



_ 1 / «., K c, \ 



which is the correct form already obtained, (5, i). 
Not only the argumentation by which the formula 

i/o2 _ f^ + b^-^c,^+ 

J/,;i ~ a.j + 6i + c, -f- 

was obtained, but also this formula itself, is wrong. By saying, 
a formula is wrong we mean it is not universally right, although 
it may be right on some occasions. This formula is wrong be- 
cause it does not universally carry out — and in practice is rarely 
11 



162 MATHEMATICAL FORMULATION 

likely to carry out — the idea we intended to carry out, namely 
to employ even weighting. For (as is more fully shown in 
Appendix A, II. § 7) this formula really contains a special 
system of weighting in the case of averaging the variations, as 
it agrees with the formula 

Mo2 1 / a„ , 6. c. 



Mo, 



la,— + 6, 7^ + c,— + to n terms), 

V \ 'b^ 'c^ J 



(in which n"= a^ -\- b^ -\- c^ -\- to n terms), which is the 

formula for averaging the variations with the weights a^ for [A] , 
6j for [B], Cj for [C], (so that there would be even weight- 
ing here only in case it happened that a^ := 6^ = c^ = ). 

But we have probably chosen these terms at haphazard, that is, 
without paying regard to the importance of the classes : we 
have sought merely, in the different classes, of the usually em- 
ployed mass-units the numbers that happen at the first period 
to be equivalent to the money -unit. The weighting for the vari- 
ations which we are really using is, then, haphazard weighting, 
and the whole process is absurd. 

No reliable result can be obtained in this way, and — which 
proves it — an mdefinite number of different results may be so 
obtained. This may be shown by an example. Suppose the 
money-unit purchases two quarters of wheat at the first period 
and at the second three, at the first four bales of cotton and at 
the second two, at the first five tons of coal and at the second 
still five ; and suppose we are trying to measure the exchange- 
value of money m these three other things only. We might then 
be tempted to represent the average exchange- value of money at 
each period, and consequently the variation of the averages, in 
the following way, 

if,,_ K3 + 2-H5) _ 3 + 2 + 5 _ 10 

3Io, - i{2 + 4 + 5) - 2 + 4 + 5 " IT " ^•'^^'^' 

which indicates a fall of 9.09 per cent. Now if it had happened 
that wheat had been measured in bushels instead of in quarters, 
the exchange-value of the money-unit at the first period would 
have been equal to that of sixteen bushels of wheat and at the 



POSSIBLP] ERRORS 163 

second to that of twenty-four. Then if these figures had been 
used, we might have represented the averages at each period 
and the comparison as follows, 

Mo,^ i(24+^2 + 5) ^ 24 + 2 + o _ 31 _ 
Mo, Kl<^+'4 + 5) 16 + 4 + 5 25 -^' 

which indicates a rise of 24 per cent. 

The reason for the difference in the results is plain. It is 
that in the latter case a great deal more importance or weight is 
attached to the rise of the Avheat than to the fall of the cotton 
and to the constancy of the coal, whereas in the former case 
there was more importance attached both to the fall of the cot- 
ton and to the constancy of the coal. Now if even weighting 
were employed, or intended, the proper measurement would be 
in this form, 

Jfo2 1/3 2 5\ 1/24 2 



M,n 



1/3 2 5\ 1/24 2 5\ ^ 
= 3(2 + 4 + 5J=3(l6 + 4 + 5J=l' 



which indicates constancy (on the supposition that the arithmetic 
average is the right one to employ, the rise of wheat by fifty 
per cent, being counterbalanced by the fall of cotton by fifty 
per cent.). The same result would be obtained if we put the 
comparison in this form 

Mo. _ Ki K^]+K4]+i[5] ) 

Morki^ [2]+l[4]+l[5])' 

and used only the outer coefficients. This is what we did when 

we employed the algebraic symbols a'a, b'b, c'c, for the 

second period and a, b, c, for the first, and then counted 

only a', b' , c',. . This method gives only one result — the 

only result which can be given Avith the employment of even 
weighting for the variations. The other gives as many results 
as there are mass-units that might be employed. That these 
various results are given by employing variously weighted 
averages of the variations, in which the weighting is according 
to the number of mass-units purchasable at the first period, 
may be shown in the above instances as follows. 



164 MATHEMATICAL FORMULATION 






Jfo2 _ 3 + 2 + 5 1 /^ 3 . , 2 
and 



Mq, 24 + 2 + 5 1 / 2^ 2 5\_31 

lfoi~16 + 4 + 5~25 V ^ 16 "^ ^ 4 + ^ Jj I ~ 25 



Thus, in using the arithmetic average, when we want to em- 
ploy even weighting, we must divide the exchange-vakies of 
the money-unit in each class at the second period by its ex- 
change-value in the same class at the first period, then add the 
quotients, and divide by the number of the classes. This method 
may, for short, be called the method of dividing before adding. 
The other would be the method of adding before dividing. The 
latter always gives the same result as the former, and correctly 
carries out the intention, if we first reduce the numbers of the 
mass-units purchasable at the first period to the same figure, pre- 
ferably to unity, or to 100 (by creating, so to speak, a special 
mass-unit for each class). It is wrong simply to add up the pur- 
chasable numbers of any mass-units that happen to be employed. 
In doing so, we are really using uneven weighting for the classes 
according to the numbers of these units which happen to be pur- 
chasable at the first period. The larger the mass-unit which 
happens to be employed, the smaller is the number of these units 
purchasable, and the smaller the weighting ; and the smaller the 
unit, the larger the weighting. Yet in spite of itself, it is con- 
ceivable that this method might happen to be right. For, 
though intending to employ even weighting, if things so hap- 
pened that the uneven weighting really involved should be the 
correct weighting, then our operation would be much better than 
what we intended. But this would happen only on the very 
improbable supposition that the numbers of the mass-units of the 
different classes purchasable at the first period were directly pro- 
portional to the sizes of the classes properly estimated. It is es- 
pecially unlikely to happen because the custom is to employ larger 
units for the more important classes, so that the Aveighting is 
more apt to be the opposite of what it ought to be. If we em- 
ployed the same mass-unit throughout, for instance, a pound, the 



POSSIBLE ERRORS 165 

weighting would be inversely according to the preciousness '^ of 
the classes, — which would mostly be in the right direction, but 
enormously exaggerated in some instances, and generally most 
irregular. 

This employment of a single mass-unit seems to have the 
recommendation that it does away with the multiplicity of 
results obtainable, and so with our perplexity about deciding 
between them. But it is not so ; for if we employed a weight- 
unit throughout, or a capacity-unit throughout, the results in the 
averages of each period separately and in comparison would be 
different. Evidently neither measure is more authoritative than 
the other, and neither provides the weighting which any rational 
treatment of the subject would recommend. 

It is, however, questionable whether in practice any of these 
methods is worse than the method carrying out even weighting 
— that is, whether the weighting in any of these methods de- 
parts more from the true Aveighting than does even weighting. 

§ 4. If, then, we wish to employ proper weighting, and think 
that we have found what it ought to be, representing the weight 
for [A] by a, that for [B] by b, that for [C] by C, and so on ; 
if we proceeded as before and introduced this weighting for the 
classes in the averages at the periods separately, and then com- 
pared them, as represented in this formula 

3Io, a(«,.4) + b(6>) + c{c^C) -h 



and then dropped the A, B, C, , wrongly taking them for 

units, or as merely referring to the classes, and so used the 
formula 

Mq, a «2 -I- b 62 4- C C2 + 



Mor aai-hb6, -hCCj-h 

our formula would still be wrong, and would still not universally 
do what we intended ; for now the weights of the variations 
would be (as shown in Appendix A, I. § 8), not a, b, c, , as 



^ Preciousness is an idea in economics very much like density in physics. 
Density is the ratio of the weight of a body to its volume. Preciousness is the 
ratio of the exchange-value of a body either to its volume or to its weight. 



166 MATHEMATICAL FORMULATION 

intended, but aa^, b 6^ cc^, , that is, something else, irreg- 
ular and haphazard, containing the same accidental factors as 
before, namely, the quantities at the first period purchasable 
with one money-unit and measured in various mass-units. And 
there would also be an indefinite number of results obtainable, 
according to the possible combinations of possible mass-units 
that may be used. But, of course, if the reduction had been 

made so that a^, b^, c^, are units, or any uniform quantit}^, 

then the weighting employed would be the one intended for the 
variations. Naturally, with the proper weightmg rightly ap- 
plied there can be only one result. 

Thus in our previous numerical example, suppose at each 
period (or on the whole) we found the class wheat to be twice 
as important as the class cotton and three times as important as 
the class coal, and wanted to use this weighting. This weightmg 
for the variations would be carried out in the following manner, 

|j = i(3x| + 2x| + lx|) = 1.08J, 

indicating a rise of 8 J per cent. But if we employed the same 
weighting in the method of averaging separately on the unequal 
figures above instanced, we should have in the one case 

Mq, _ i (3 X 3 + 2 X 2 + 1 X 5) _ 9-^4 + 5 _ 1 8 _ 

^i"oi ~ K3 X 2 + 2 X 4 + 1 X 5) ~ 6 -h 8 -h 5 ~ 19 '^ '' 

indicating a fall of 5.3 per cent. ; and in the other, 

Mq, 1(3x24 + 2x2 + 1x5) 72 + 4 -f- 5 81_ 

Mo, K3xl6 + 2x4 + l X 5)~48 + 8 + 5 61 ' ^'' 

indicating a rise of 32.7 per cent. These discrepancies are due 
to the fact that in the first operation, which carries out the 
weighting intended, we are attaching slightly more importance 
to the rise of wheat than to the fall of cotton and to the con- 
stancy of coal ; in the second, we are really attaching slightly 
more importance to the fall of cotton (in the proportion of 8 to 
6 and 5); in the third we are really attaching much more im- 
portance than we intended to the rise of wheat (in the propor- 



I'ossiblp: EHiJoits 1()7 

tion of 48 to H and 5). When we eome to treat of tlie measure- 
ment of the exchange-vahie of money by means of prices, we 
shall find a convenient way by which the right result may in 
some cases be very conveniently reached by the use of this 
method of adding before dividing. But here no such method is 
forthcoming. For the operation represented in the formula 

Mq, _ 1(3 X 1| [2] +2 X K-^] +1 X 1 [5]) 
^« ~ K3 X 1 [2] + 2 X 1 [4] + 1 X 1 [5]) 

has nothing to recommend it, as its numerator performs the 
whole operation of the method of dividing before adding, and 
its denominator is pure waste, reducing to unity and not affect- 
ing the result. 

§ 5. Harmonic averaging. — Intending to use even weighting, 
we might be led to average each period and to compare the 
averages in this way, 

1 

1 f 1 i i 



31,, 1 



1 f 1 1 1 

+ 77-75^ + TTT^ + 



71 



{a,A)^{b^B) -^{0,0) 



and if, as before, we incorrectly eliminated A, B, C, , we 

should liave, after reducing and transposing, 

1 1 1 

,. - + r + - + 

31o2 «i '\ t'l 



3Ior~l I I 

a+K+7 + 

And here, too, not only the reasoning by which this result is 
reached, but this result itself, is wrong. For (as is shown in Ap- 
pendix A, III. § 7) this is the formula for the harmonic aver- 
aging of the variations with weighting of — for [A] , of j- for 

[B] , of — for [C] , and so on, that is, with weighting inversely 
according to the sizes of the figures at the first period. Hence 



168 MATHEMATICAL FORMULATION 

a multiplicity of results would be obtained according to the 
mass-units we happen to employ. Here the larger the mass-unit, 
the smaller the number of it purchasable with the money-unit at 
the first period, and the greater the influence of the variation ; 
and the smaller the mass-unit, the smaller the weight. And 
here, too, it is conceivable that such even weighting in the sepa- 
rate averages, giving uneven weighting in the average of the 
variations, might give the right uneven weighting, although this 
is exceeding unlikely to occur. It is not so unlikely, however, 
as in the preceding case ; for we have seen that the weighting 
there was likely to be the opposite of what it ought to be. 
Here the weighting, being the inverse of that, is therefore likely 
to run more in the right direction, although, of course, only in 
an irregular way. On the other hand, the employment here of 
the same mass-unit throughout would tend to make the error 
run in the wrong direction. 

If, instead, we wished to employ proper weightuig, and, finding 
it the same at both periods, introduced it in the separate aver- 
ages, and compared them in this form. 



1 f a b c 



Mo, 1 



1 r a b c 



and in the same manner reduced this to 

a b c 

i^T - + r + - + 

Mo2 «! Oj c^ 



Ma^ a b c ' 

«2 ^2 ^2 

we should again be performing a wrong operation, as the weight- 
ing in this formula, for the variations (as shown in Appendix A, 

a b c 
III. § 8) is not a, b, c, as intended, but — , -j-, — , , and 

Oj, O, C-j 

the correct weighting is again perverted by a haphazard factor. 



POSSIBLE p:rr()rs 109 

It is not worth while to give numerical examples here, or to 
expatiate further on the errors incurred through negligence in 
the treatment of harmonic averages, although this is the average 
which has been mostly adopted in measuring the exchange- value 
of money. But it has generally been adopted from the 0})posite 
view-point of the measurement of price-variations by means, as 
we shall see presently, of the arithmetic average, in treating of 
which we shall have to revert to this subject. 

§ (). Geometric averaging. — Here everything is simpler. The 
weighting in the comparison of the averages at the two periods 
and the weighting in the averaging of the variations, if appar- 
ently the same, are so really ; for it does not now matter whether 
we multiply before dividing or divide before multiplying. If 
we employ even weighting in the one case, we have even weight- 
ing in the other, as is hereby shown. 



Mg.^ v'^a^-b.^-C^- ■■■ 


-=\y 






^2 


Mqx v^a^-b^-G^- - 




Cl 



And if we employ the same uneven weighting in each separate 
average and in the average of the variations, we really have the 
same weighting in both cases. 

The wrong reasoning by which the erroneous results were 
reached in the first two cases has no room here. In reducing 



Mo. v{a,A).{b,B).{c,C). 



Mor y{a^A).{bfi).{c^C) 

to the above expressions by dropping A, B, C, from the 

two sides of the fraction, we have a perfect right to do so ; for 
although these expressions are not units, they are the same on 
both sides. The whole expression cannot be further simplified 
by dropping the radical signs, because the products are not in 
the same ratio as their roots. 

Now the same apparent weighting being the same real weight- 
ing in this kind of averaging, a warning is still needed. Aver- 
aging the periods separately, we might be inclined to weight 
every figure according to the number of times it occurs, what- 



170 MATHEMATICAL FORMULATION 

ever the size of its class may be ; and if these numbers happen 
to be the same at both periods, the comparison of the averages 
would be 

Mq.2 v'^ ci^ -^2 '^2 ^ ^ terms 

^01 v'^ a^ ■ h^ . G^ to n terms 

in which the exponents x, y, z, refer to the numbers of the 

mass-units that appear in trade, and n' = x -{- y -^ z -f to n 

terms. We might then be inclined to adopt this weighting into 
the method of averaging the variations. 

This would be wrong. The results would be various according 
to the sizes of the mass-units accidentally chosen. For, the 
larger the mass-miit chosen, say, for [A] , the smaller would be, 
not only a^ and a^, (the numbers of times the mass-unit is pur- 
chasable with one money-unit, which are here immaterial, since 
their proportion is always the same), but also x (the number of 
times the mass-unit apjDcars in trade), that is, the weight of this 
variation. Therefore this weighting would be haphazard. If, 
in order to avoid the multiplicity of results which would ensue, 
we adopted only one mass-unit throughout, the weighting would 
be influenced by the relative preciousness of the articles ; and, 
as before, as preciousness may be measured in two ways (by 
comparison of the exchange-value either with weight or with 
bulk), there would still be two distinct acts of weighting. The 
proper weighting is that which we have already discovered, — 
the number of tunes, not the accidentally actual, but the real (or 
ideal) economic units occur in each class. This weighting should 
be employed in the averaging of the variations ; and if we happen 
to prefer, for any reason, the other method of averaging each 
period separately and comparing their results, we ought to em- 
ploy the same weighting there. 

§ 7. One more point deserves to be considered here. We 
have seen that the errors we have been warning against we 
shoidd have been apt to slip into had we employed the method 



POSSIBLE ERRORS 171 

of notating the quantities at the second period independently, 
without expressing their relation to the quantities at the first 
period. We should have been especially apt to fall into them 
had we additionally employed the imperfect method first noticed 
of formulating exchange-value relations by means of the equi\-- 
alences of mass-quantities. For then — we may here confine 
our attention to the arithmetic method — from the formula for the 
first period, 

Mj '== a^A ==b^B === CjC =-= to n terms, 

we might have deduced 

Mj === - (ffjA + 6,B -f CjC -I- to n terms) ; 

and from the similar formula for the second period we might 
have deduced a similar formula of the combination, and then 
the comparison would be in this form, 

^ - (a^A 4- 62B -(- c^C -|- to n terms) 

M ^^ i ' 

^ - («jA -I- &jB + CjC -\- to n terms) 

in which as the terms A, B, C, merely refer to mass-units 

of the different classes, and so represent units, we might — es- 
pecially if only one mass-unit had been employed for all the 
classes — have thought ourselves justified in omitting them and 
reducing the formula to this, 

M, a, -h 6, -h c„ -(- 



• M/"" a^ -h 61 + Ci -f- 

— the fiiulty formula already treated of. Here, too, however, 
our reasoning would have been wrong, as is obvious if A, B, 
C, referred to different mass-units. But even if they re- 
ferred all to the same mass-unit, our reasoning would have been 
wrong, as we are not dealing with quantities of weight or of bulk, 
but with quantities of exchange- value (namely the exchange- 
values of certain weights or volumes of things). Here, for our 
purpose, a pound of lead is very different from a pound of feath- 



172 MATHEMATICAL FORMULATIOX 

ers. It is only a dollar's worth of lead that is worth as much 
as a dollar's worth of feathers. 

§ 8. This kind of formulation calls for some remarks. The 
arithmetic formula 

M =0= _ (aA + 6B + cC + to n terms) 

n 

may be read : " The money-unit (or anything else) is equiva- 
lent to the sum of the quantities of the thiugs it can exchange 
for or purchase, divided by the number of their kinds"; or 
again : " The purchasing power of the money-unit (or of any- 
thing else) is the power of purchasing the sum of the quantities 
of the things it can purchase, divided by the number of their 
kinds," the supposition being that the kinds or classes are 
equally important. And similarly if we used this formulation 
for the other two kinds of averages. The peculiarity of this 
formulation, and of such interpretations of it, is that there is ref- 
erence only to the mass-quantities of the things purchasable, 
and no reference to the exchange-values of these masses, or of 
money in their classes. Evidently such an average would have 
manifold results according to the sizes of the mass-units em- 
ployed. Hence some method has to be added of selecting the 
mass-units. One is to use always the same mass-unit — either a 
weight-unit, or a capacity-unit. With the addition of such a 
method (the difference between the two not generally being no- 
ticed), such is the employment often made of the term " purchas- 
ing power." Yet not only this formulation does not yield a 
mathematically serviceable formula, but also these interpretations 
of it, with the added methods of selecting the mass-unit just no- 
ticed, and the term " general purchasing power " so used with 
reference merely to the mass-quantities of the things purchas- 
able, do not yield clear ideas. The things purchasable whose 
mass-quantities alone are thus united and divided, or averaged, 
form only a promiscuous conglomeration of variously valuable 
things with many and great qualitative distinctions, which are 
entirely ignored. The partial similarity in this method of meas- 
uring general purchasing power with the true method of meas- 



OF PRICE RELATIONS 173 

uring particular purchasing powers, and the complete erroneous- 
ness of it, will be more fully ])ointecl out in a later Chapter. 

It is possible, however, to put a true interpretation upon the 
formulation by equivalence. Tlie above-given arithmetical aver- 
age may be made to read : " The general purchasing power of 
the money-unit (or of anything else) over all other things is 
equal to the sura of its particular purchasing powers ovei' every- 
thing separately, divided by their number " — under the supposi- 
tion of even importance. But here we merely have exchange- 
value under another name, based upon a formula mathematically 
imperfect. We must prefer, therefore, not only the other kind 
of formulation, but also the term " exchange- value." It is a 
canon in logic that we should avoid the use of two terms with 
identical meaning, and should choose the term which more clearly 
expresses the meaning.'' 

IV. 

§ 1. Thus far we have made no use of prices. Yet in all 
monetary matters the use of prices is a great help. Almost all 
the attempted measurements of the exchange-value of money 
have proceeded by drawing averages of prices. We must there- 
fore restate the previous procedures in terms of prices. 

We have been dealing directly with the exchange- value of 
money in other things. Prices express the exchange- value of 
the other things in money. Therefore the variations of ])rices 
are the inverse of the variations of the exchange-values of money 
in the things priced. And a variation of an average of the 
variations of jVrices will indicate the inverse variation of an 
average of the variations of the exchange-values of money in the 
things priced. Hence the possibility of substituting measure- 

* We can think of the " lifting power " of a derrick without thinking of a lift- 
ing power in the things it lifts. And so we can think of the " purchasing power " 
of money without thinking of the purchasing power of the things it purchases. 
In fact, this term "to purchase" always means that we give money for some- 
thing. Hence money actually has purchasing power, in this proper sense, with- 
out anything else having purchasing power ; and so " purchasing power " is not 
a correlative term. But we cannot employ the term " exchange-value " without 
the idea of exchange, which involves correlative exchanges, and consequently 
correlative exchange-values. 



174 MATHEMATICAL FORMULATION 

ments of prices for direct measurements of the exchange-value 
of money in other things. 

If we start with mass-quantities equivalent to the money- 
unit, and with prices therefore at unity, the subsequent variations 
of the exchange- values of the money-unit and of prices will be, 
as we have seen (in Proposition X.), to reciprocals of each other. 
Now we commenced our previous enquiries always by supposing 

M<:=aA=c=6B-=cC=^ , which means that the prices of a A, 

of 6B, of c C, are one money-unit each (whence the price of 

one mass-unit of [ A] , namely A, is — of B y, of C ^ , and so 

on). Therefore, at the first period, if we draw the averages of the 
prices of these equivalent mass-quantities, however large or small 
these be, we shall always get unity, as before. And at the second 
period, when M=c= a'a A o6'6 B =0= c'cC = , the prices of 

a A, of 6 B, of c C are ,, p, —,, Now letting P^ and 

P, represent the averages of the prices at the first and second 
periods respectively, as Pj in every case is 1, in every case 

P, 

-p^ = P^, that is, the comparison of the two averages is the same 

1 
as the average at the second period. Therefore, again supposing 

that we are dealing with classes equally important at each 

period, or over both the periods together, by inverting the order 

and placing the harmonic average first and the arithmetic second, 

for a reason which will appear immediately, we have 



l(a' + 6' + c'-h )(12,.) 



(12,0 
(12, 3) 



OF PRICE RELATIONS 175 

But from the exchange-value comparisons we had (4, i-s) 

Mo, 1 

Mo, 



n 

Mo, 
To. 



n\a' ~^ b' "^c' + ) 






Thus it is evident that in the order given the first formuki for 

P 

:p^ yields a result the reciprocal of the result of the first formula 

M o P., 

for Yr^ } ^liG second formula for p' a result the reciprocal of the 

M ., 

result of the second formula for ,^ , and the third formula for 

Mo,' 
p J^/ ^ 

^ a result the reciprocal likewise of the third formula for ^^' . 

Now the first formula here expresses the harmonic average of 
the prices at the second peoiod (or of the variations of the prices), 
while the first formula there expressed the arithmetic aver- 
age of the variations of the money-unit's exchange-values ; 
the second formula here the arithmetic average, the second there 
the harmonic ; but the third here and the third there both ex- 
press the geometric average. Thus when the harmonic average 
of prices shows a rise of the exchange-values of other things in 
money by p per cent, to 1 -\- p times their former level, which 
means a fall of the exchange-value of money in the things 

1 1 p 

priced to :, by 1 — z. or :, per cent., this fall is in- 

^ 1 -i-p ^ 1 -j-p I -\-p '- 

dicated by the arithmetic average of the exchange-values of the 
money-unit in those other things ; and reversely if the former 
shows a fall. When the arithmetiG average of prices shows such 
a rise of the exchange-values of other things in money, which 
means such a fall of the exchange-value of money in the other 
things, this fall is indicated by the harmonic average of the ex- 
change-values of the money-unit ; and reversely if the former 



176 MATHEMATICAL FORMULATION 

shows a fall. But when the geometric average of prices shows 
such a rise of the exchange- values of other things in money, which 
means such a fall of the exchange- value of money in other things, 
this fall is indicated also by the geometric average of the ex- 
change-values of the money unit ; and reversely if the former 
shows a rise. Agam, if on one set of variations the harmonic 
average of prices shows constancy of the average of the exchange- 
values of other things in money, which means also constancy in 
the average of the exchange-values of money in other things, 
this result is, on the same variations, indicated by the arithmetic 
average of the exchange-values of the money-unit. If on an- 
other set of variations, the arithmetic average of prices shows such 
constancy, it is indicated, on this set of variations, by the har- 
monic average of the exchange-values of the money-unit. But 
if, on still another set of variations, the geometric average of 
prices shows such constancy, it is indicated, on the same set, 
likewise by the geometric average of the exchange- values of the 
money-unit. 

This relationship between these averages is in accordance with 
a well-known mathematical theorem concerning these averages. 
The harmonic average of any given quantities is the reciprocal 
of the arithmetic average of the reciprocals of those quantities. 
Reversely the arithmetic average of any given quantities is the 
reciprocal of the harmonic average of the reciprocals of those 
quantities. But the geometric average of any given quantities 
is the reciprocal of the geometric average of their reciprocals. It 
is precisely with such quantities and their reciprocals that we 
are dealing. 

We can therefore find the average variation of the exchange- 
values of money in all other things by drawing the average 
variation of the prices of all things, but noticing that the har- 
monic averages of prices is the one which gives the inverse of 
the arithmetic average of the exchange-values of money, the 
arithmetic average of prices the one which gives the inverse of 
the harmonic average of the exchange-values, while the geo- 
metric average of prices corresponds also with the geometric 
average of the exchange- values. 



OF PRICE RELATIONS 177 

The problem still remains as to which — if any — of these 
averages is the right one. The proper order of the enquiry 
would seem to be as to which is the right average to express the 
variation of the exchange-values of the money-unit in all other 
things. If we find this to be the arithmetic average, we shall 
know that the proper average to draw of prices is the harmonic. 
If the right average for the exchange-values turns out to be the 
harmonic, then, and only then, will the arithmetic average of 
prices be the right one. But if the right average in the one 
case is found to be the geometric, the geometric will also be the 
right average in the other. It would seem that to turn the 
problem around and to start with, or to confine ourselves to, an 
enquiry concerning the proper averages of prices would give us 
a wrong point of view and put us at a disadvantage. 

§ 2. The formulae for the averages of prices, of course, are 
not to be confined to the form in which the prices at the second 
period are stated as reciprocals of the variations of the exchange- 
values of the money-unit in the things priced, that is, in the 

fractional forms — , t^, — , We may give them upright 

integral forms. Let us employ the accented Greek letters, a' , 

/9', J-', , to rej^resent the prices of « A, 6 B, eC, at the 

second period — these being the quantities which at the first 
period were all priced at 1.00, so that their prices at the second 
period directly express the variations of the prices. Then, as 

a! replaces — , /9' 77, •/ — , in the preceding formulae, we 

have the following : 

P„ 1 



Pi 1 /I 1 1 



/111 \' 

t^?V^ ) 



(13,0 



' = k«'+/5' + r' + ), (13,.) 



Pj ft 



p 
12 



'^ = va'.Ci>.y' (13,3) 



178 MATHEMATICAL FORMULATION 

We can also employ the other method of notation. We may 
represent the prices of any mass-nnits of the various classes at 

the first period by a^, ^^, y^, , and the prices of the same 

quantities of the same classes at the second period by o.,^, /J^, y.^ 
Then the variation in the preceding formulae represented 

by a/ is here represented by -, and that represented there by 

/9' is here represented by ^, and so on. And the preceding 
formulse become the following : 



. ^ + 5^ + ~^ + 



P.-.A«.+r^ >' ''''"^ 



Pi «i ^1 'i\ 



2^^/s./^.r2 (14,3) 



In these formula? the terms are the reciprocals of the terms in 
formulae 5, 1-3, that is, — = —,^ = / , -^ = -' , and so on, and 

^ p, , 

inversely. Hence each formula here expresses for p'therecip- 

31 ., 
rocal of the formula there given, in the same order, for -^ ■ 

It is, of course, more natural to use the prices of the custo- 
marily employed mass-units of the diflFerent kinds of goods ; and 
it is more convenient. In practice all reductions of prices at the 
first period are labor wasted. 

Y. 

§ 1. The mere introduction of uneven weighting into these 
formulae is likewise easy. The weighting is, of course, the same 

1 These three formulae, confined to two terms, were first given by Jevons, B. 
23, p. 121, and then, in the more general form, by Walras, B. 69, pp. 12-13, B. 70, 
p. 432. 



OF I'lUCK DELATIONS 179 

as in the measurement of the exchang-e-vahies of money in the 
other classes. The weighting' of the arithmetic average of the 
variations there will come into the harmonic average of the 
variations here as it there entered the harmonic average ; and 
reversely. The formuUp, then, when we are justified in using, 
or when Ave do use, single weighting, will be 



rr \ - + 37 + - + to n terms 

1^ = 1 (ar/ + b,9' + C-/ + to n terms), (15, 2) 

& = \/ a'^^i'"^ J'" to ri terms, (15, 3) 

1 

in which ;/" = a -f b + C + to *i terms ; or again 

-- = -. ^ r; (1<^; ') 

^1 _ I a-l + b '-• + C -^' + to )( terms ) 

n"\ «, ^■i.^_ y.^ ) 

5^ = -l/a- + ^- + C^+ ^'' " terms Y (16, 2) 

The same comments are suggested by these formulae as by those 
for the exchange-values of money. It is indifferent how large 

or small, integi-al or fractional, the numbers a, b, C, be, 

so long as they are in the proper proportions to one another. And 
a change in the weighting without a variation in the prices (or 
with a uniform variation of all prices) produces no change in 
the result (in agreement with Propositions XLIV. and XLY.). 

1 These formula for the unevenly weighted averages have rarely been stated. 
Westergaard gives the arithmetic formula in a cumbrous form (see Appendix C, 
III. § 1) ; and adds that a " similar alteration " may be given to the other simple 
averages, but he does not give it, and the expression "similar" is misleading. 
They are unnoticed by all the other writers cited in the Bibliography except 
Edgeworth, whogives thethird in its logarithmic form, asnoticed belowin NoteS. 



180 MATHEMATICAL FORMULATION 

^e may here add that these and all the preceding proper for- 
mulae also carry out Proposition XXXVI.^ 

With high numbers for the weights the third of these formulae 
might seem unworkable. Indeed it would be unworkable even 
with small numbers for the weights, or with all the weights 
alike, — since their sum, when many classes are dealt with, would 
involve the extraction of a large root, — ^but for the help which 
may be rendered by logarithms. With the aid of logarithms 
the third formula is no longer difficult to execute,' even with 
large, or with minutely fractional, figures for the weights. This 
use of logarithms was made by Jevons, although he employed 
only even weighting. With uneven weighting, the logarithm 
of the result may be obtained in several ways, among which the 
following may be noted : 

(1) log p^ = ^ (^a • log ^^ + b ■ log y + to n termsY 

P 1 

(2) log jr- = ^ {(a -log «, + b-log /^^ + to n terms)— 

(a • log «j + b • log /9j -f- to n terms)}, 

(3) log p^ = :^ {a (log «, - log aj + b (log ^,- log ^,) + 

to n terms}. 

The first is the ojDcration we should naturally adopt if we had 
already reduced the prices at the first period to 1.00 (or to 100), 

— then being replaced by a' already ascertained, and so on. 

The second would be the simplest, if the weighting were even, 
the expressions a, b, C, then dropping out.^ The third is 

2 This and a remark in Appendix A, I. § 10 end, deserve attention. They 
show that when we know approximately the variation of the average, it is 
more important that we should seek to be accurate with the weighting of the 
classes whose variations are greatly divergent from that of the average than with 
the weighting of the classes whose variations are nearly the same as that of the 
average. See also Note 21 in Chapter IV., Sec. V., § 10. 

3 Very nearly in tliis form, with even weighting, the formula is given by 
Walras, B. 61, p. 7. A formula very nearly in the first form, with uneven weight- 
ing, has since been given by Edgeworth, B. 59, p. 287. 



OF PRICE RELATIONS 181 

generally the most useful operation, saving the divisions needed 
in the first and the doubling of the multiplications in the second.* 
§ 2. We may, of course, here as well as in the direct meas- 
urement of the exchange-value of money, need to use double 
weighting. The formulae embodying double weighting in the 
case of prices may be directly derived from the formulae pre- 
viously given for double weighting in the other case (formulae 
11, 1-3), We must still employ some expressions — x^, x.^, y^, y.^, 
etc. — to represent the numbers that are produced or consumed 
at each period of the mass-units whose prices are given. 

And we must remember that we must make two inversions, first 

p J/ 
of the numerators and denominators, for ~- = ^r — j ^^^ second 

of the exchange-value symbols and the price symbols, for 

ct o. 

— = — and reverselv, and so on. Then those formulae become 

these, in the same order : 

j^ "71 - + 1^ + - + to « terms ) 



* 1 (x. v.. z., \' 

' —,\-+4+~+ to n terms ) 

^«2 \«2 P2 r-i J 



(17,1) 



P --/ (■'^■2«2 + y-A + ^^2/'2 + to « terms) 

-2 = ^h , (] 7, 2) 

PI ' V ' ^ 

' —t G'^i«i + VA + ^iTx + to 71 terms) 



T3 ?'^/^«/^ ■ ^o'^ ■ To"''^ ' to n terms 

£2 ^ 1/ ' ^' '' n 7, 3) 

?! r^/'fii'^i • A^i • Ti'- tow terms 

The anticipatory remark may here be made that in the future 
course of these pages we shall find no use for any of these 
formulae except the second, which is the general formula for 
double weighting in the arithmetic average of prices. This 
second formula may be expanded in full as follows, 

■* Jevons used three-place tables. It would be advisable, however, to use tables 
with not less than six places. 



182 MATHEMATICAL FOEMULATIOX 



•V^-2+ ^2^'2-^- V/2+ 




Po ^2 + 2/2 + ^2 + 




Pi --^-i"! + yA + hTx + ' 




•^'1 + i/l + ^1 + 

iiices to 

x./x.^ + y-ii^-i + ~2r2 + -^'i + i/i + --i 


. + 


•^'i«i + yA + ^in + -^2 + 3/2 + '^2 


, + 



0--IM 



, (17,. 3) 
1 •''I'^i -r ^i/^-i -r -^i/i -r -^2 ~r U2~r ■--2 "t 

this last being the best form for it. 

Here, as before, it is evident that in all these formulae the re- 
sults will be variously aifectecl by the sizes of the mass-units we 
happen to choose. Here also, therefore, a method of selecting 
the mass-units must be established. And this method need not, 
or it may be, indicated in the formula itself. It is not indicated 
in the formula used by Drobisch, who therefore employed the 
simple formula here given, after first telling how the mass-units 
are to be chosen. It is indicated in the formula used by Pro- 
fessor Lehr, Avhich therefore becomes more complex, though the 
real difference between his method and Drobisch's is not in the 
formula, but in the method of choosing the mass-units, and his 
method is not so difficult of application as Drobisch's. Both 
these methods use the second formula, which admits of other 
variations through the employment of several different methods 
of selectmg the mass-units. 

In this second formula there is the peculiarity that the sizes 

of the mass-units have no effect upon the terms x^a^, x.^a^, etc., 

that is, upon any of the terms in the first half of the formula 

in the form last given. For the larger the mass-units, the larger 

will be a in the same proportion, and x will be smaller in the 

inverse proportion, so that their product, xa., will be unaffected, 

and so in all the other classes, and at both periods.^ Hence in 

^E. g., if [B] be wheat aud y-^ represent a thousand quarters, then /3i represents 
the price of a quarter — let us say eight dollar's, wherefore i/i /3i = 1000 X 8 = 8000. 
If instead we used bushels, 2/1 would represent eight thousand bushels, but (i-i 
would represent one dollar, so that we should still have y^ /S^ = 8000 X 1 = 8000. — 
This peculiarity exists in the corresponding second formula for the exchange- 
value of money (11, 2), expressing the harmonic average ; for there no influence is 

exerted upon the terms ~ , etc., since the larger is ?/i, the larger also is b^ in the 

61 
same proportion, and the ratio remains the same. Thus in the above example 
1000 _ 8000 
1/8 1 



OF pricp: helations 1<s;} 

this form of this formula the first half remains always the same, 
and the modifications, if made at all, are to be made only in the 
second half. 

This formula may be simplified. In the first half the whole 
denominator is merely the total valuation of all the goods pro- 
duced and consumed in the first period at the prices of the first 
period ; and the whole numerator is a similar total valuation for 
the second period. Naturally these parts of the formula remain 
unchanged whatever be the mass-units employed. In the second 
half the whole numerator is the simple inventory of all the mass- 
units of the goods produced or consumed in the first period ; and 
the whole denominator is the similar inventory for the second 
period. Naturally these parts are aflPected by the sizes of the 
mass-units in which these numbers are reckoned. Now the jiarts 
in the first half represent the total money-values of the goods at 
the first and second periods, and hence, as wholes, may be repre- 
sented by V, and V., ; and the parts in the second half repre- 
sent the total mass-quantities of the goods at the two periods 
respectively, and hence, as wholes, may be represented by Q, 
and Q.,. Then 

"& = ^.^ (17...) 

is a simplified form of the second formula. 

The real nature of this second formula is indicated in the 
second form in which it has above been ])ut, which may be sim- 
plified into this, 

v., 

^ = ^. (17^.) 

For here it is seen that the formula is merely that of a compari- 
son between an average price of all goods at each period. For 
at each period the total money-value of all goods collectively is 
divided by the total quantity of all the goods ; which is the 
plain method of (arithmetically) averaging the prices of all goods 
at each period. But the total " quantity " of all the goods is 



184 MATHEMATICAL FORMULATION 

the total sum of the numbers of the mass-units of all the goods, 
which sum is affected by the sizes of the mass-units chosen to 
measure the goods by, so that the averages are arbitrary until 
the proper method of selecting the mass-units is found. This 
fact of its representing a comparison between two ordinary 
(arithmetic) averages is what gives this second formula its su- 
perior recommendation over the other two. 

It might be thought that there would be a similar recom- 
mendation for the first formula here, because it corresponds to 
the first formula for the exchange- value of money, in which the 
separate averages are the arithmetic. But in (arithmetically) 
averaging at each period the exchange-value of money in all 
goods by dividing the sum of its exchange-values in them by 
their quantity there does not seem to be so much sense as in 
(arithmetically) averaging at each period the exchange-value of 
all goods in money by dividing the sum of their exchange-values 
in money by their quantity. There is congruity bet-sveen the 
money- values of commodities and the quantity of commodities : 
there is none between the commodity- values of money and the 
quantity of commodities. 

VI. 

§ 1. The averaging of prices gives occasion for similar mis- 
carriages of intention as the averaging of the exchange-values of 
money. Jevons's assertion that there is no average of prices, 
and consequently no variation of the average of prices, must be 
regarded as erroneous, except in the sense that there is no real 
or absolute averag-e already given to us in nature. We must 
construct the average, and can do so wrongly or rightly ; where- 
fore the problem really is to find how to construct it rightly. 
In the attempt to solve this problem two methods have presented 
themselves. The one is to construct separate averages, one at 
each period, on the mass-quantities that exist at each period, 
variously reckoned according to the mass-units chosen ; where- 
upon the comparison lies between these two averages. The other 
is likewise to construct separate averages, and to compare them, 
but to use in each the same weighting — as if the same mass- 



ERRORS INCURRED 185 

quantities existed at both the periods compared. Now when- 
ever this second method is employed, it is the same as some 
method of averaging the price variations with single weighting ; 
and reversely, whenever single weighting is employed in aver- 
aging price variations, — or whenever the method of averaging 
the price variations is employed, since in this there can only be 
single weighting, — it is the same as some method of averaging 
the prices at each period se])arately and comparing the averages. 
Tlierefore the single weighting for the price variations which is 
involved in the method of separately averaging the })rices and 
of comparing the averages ought to be the same as would be the 
weighting did we avowedly pursue the method of averaging the 
price variations ; for it would be absurd to employ a method of 
comparing averages of prices that is really the same as a method 
of averaging the price variations, and to em})loy in it a system 
of weighting for the price variations which we would reject if 
we consciously employed the method of averaging the price 
variations. Thus, as before (Sect. III. § 2), we have a criterion 
that the method of coinpai-imj separate price averages witJi. the 
same weighting in each should always agree with the method of 
averaging the price variations with the proper weighting for these, 
whatever this Meighting may be ; for at all events it is often easy 
to reject flagrantly improper weighting, or weighting that has 
no reason in its favor. 

This criterion is of importance because the single weighting 
for averaging the price variations involved in comparing the 
separately drawn averages is not always the same as the weight- 
ing used in each of these averages. It is always the same if we 
construct at the first period rather a level of prices, which breaks 
up into an uneven range of prices at the second period ; for in 
this case to draw an average of these later prices is really to 
draw an average of the variations of the prices, with the same 
weighting. But if we start at the first period with various 
prices, there is generally, though not always, a divergence be- 
tween the weighting in the separate averages and the weighting 
for the price variations involved, and to the ignorant hidden, in 
the comparison of those. It is here that the opportunity for 



186 MATHEMATICAL FORMULATION 

error enters in this method of comparing averages^ systems of 
weighting for the price variations being admitted which would 
never have been allowed had their existence been perceived. To 
the errors so occasioned we may for the present mostly confine 
our attention, but also noticing an allied error in a method using 
double weighting. Again the averages need to be treated sepa- 
rately. 

§ 2. Harmomc averaging of 'prices. — This corresponds to the 
arithmetic averaging of the exchange-values of money, but its 
form is the same as the harmonic averagiug of those exchange- 
values. The possible error here is that of supposing we are 
using even weighting in using tlie following formula, 

111 

T5 - +T + - + 

?? _ ' ^1 Pi Tx 

P~l 1 1 

-+T+- + 

'h i^2 r-2 

or that we are using weightmg according to a, b, C, in using 

the following, 

a b c 

P, a b c 

«2 /^2 r-i 

— that is, in comparing the harmonic averages of any prices 

(the prices of any mass-units) at each period separately, in each 

of these averages the same even or uneven weighting being 

used. For in the first of these formulae the weighting of the 

. . . „ ,. Ill , . , 

price variations is reallv according to — , y, — , , and m the 

^h th /'i 
second it is according to — , — , — , , and in neither is the 

weighting what is intended unless a^ == ^9^ = j-^ = , that is, 

unless all prices have been reduced to the same figure at the 

first period.^ We know that «^ = , ^9^ = 7-, y^ = -, and so 

a^ Oj Cj 

on. Thus the weighting here is, in the first formula, according 
^See Appendix A, III. i^^ 7 and 8, already referred to. 



ERRORS IXCrRRKI) 187 

to (l^, 6,, Cp , and in the second according to a a,, bAj, CCj, ; 

which is the same weighting as we discovered in the correspond- 
ing forms of the arithmetic averaging of the exchange-values 
of money (abo\'e in Section III. §§ l^ and 4). 

Various complications could be added which would occur in 
comparisons of the results obtained for subsequent periods by 
comparing every one of these with the same basic period. But 
as this method has rarely been used we need not tarry over it. 

§ 3. Arithmetic averaging of j^rioes. — This corresponds to the 
harmonic averaging of the exchange-values of money, and its 
errors, while the same in form as those which may occur in 
arithmetically averaging the exchange-values, are really the 
same as those which may occur in harmonically averaging them. 
As these latter have only been lightly treated, and as this is the 
method which has mostly been used, more attention is to be paid 
to it here. 

In the comparison of the price \-ariations we can start with 
the prices of any mass-units. Hence the simplest form of error, 
and the one earliest to appear, was to think we may simply aver- 
age the prices of any mass-units of commodities at the two 
periods and compare them. Tlie operation is according to this 
formula, 

P ;,(«. + /3. + r. + ) 

_ 

' ^^K + A + n + ) 

which reduces to 

•P2 _ S + /^2 + /'2 + 

pr «i + /5i + h + •••••• ' 

so that this method consists merely in comparing the sums in 
lists of prices at two or more periods. 

This form is wrong because, as a method of averaging the 
variations of prices it contains weighting according to the sizes 
of the figures in the denominator — the prices at the first period.^ 
But as these prices have been hit upon Avithout reference to the 

^ See Appendix A, II. '^ 7, already referred to. 



188 MATHEMATICAL FORMULATION 

sizes of the classes, the Aveighting is haphazard — as ah'eady no- 
ticed in the preceding Chapter. As the prices are larger the 
larger the mass-units chosen, the weighting Avill be larger the 
larger the mass-units. This agrees with what we have found in 
the case of the improper method of comparing harmonic aver- 
ages of the exchange-values of money. Now if it be claimed 
that all we want is to compare averages of prices, and not to 
average variations, the error of this position is shown by the 
fact that this method would give as many different results as 
there are combinations of mass-units that might be employed. 
Among these the right result would be hit upon, or approached, 
only if the prices at the first period happened to be directly ac- 
cording to the sizes of the classes rightly determined. 

This haphazard method, Avhich may be named, after the writer 
who first employed it, Dutot's method (see Appendix C, I.), has 
rarely been employed completely. Yet it has frequently been 
involved in another method which was specially invented to 
avoid it, and which may be called Carli's method (see Appendix 
C, II.). This other method is to average variations with even 
weighting by reducing all the prices at the first period to a uni- 
form level ; and then, — m a variation introduced by Evelyn, — 
in order to avoid the trouble of repeating this operation for every 
comparison, the first period is used as a common basis with which 
every subsequent period is directly compared. This method car- 
ries out its intention when comparing each subsequent period with 
the basic period ; but whenever the result obtained for one sub- 
sequent period is compared with the result obtained for another, 
the comparison acquires an entirely different character. The 
comparison of the second period, for instance, with the first, 
whose prices have been reduced to unity, is in accordance with 
this formula. 



|? = P, = i«+/3/+jV + ), 



and the comparison of the third with the same first is 
P„ ^ 1 



= P3 = -« + /5/ + r/ + ); 



p. ^~ n 



wherefore the comparison of these two is as follows. 



p 



ERRORS INCURRED 189 

- « + /V + /V + ) / _^ ^ / _^ / _^ 



which may al.so be written thus, 

^_L(i,r. 

Here the averaging of the price variations is no longer with 
even weigliting : it is with uneven weighting according to the 
variations of the prices at the earher of the two periods 
compared. Consequently the result obtained in this way is not 
the same as would be the result if we compared the third period 
directly with the second, employing even weighting ; and the 
result we do get depends upon the accident as to what period we 
happen to start with as the base, and what price variations have 
taken place since. Of course the price variations have no re- 
gard for the sizes of the classes, so that the uneven weighting 
here is purely fortuitous. To be sure, by the time of the second 
period, the variations may not be large, so that the weighting 
will not much depart from the even weighting intended. But 
the further we go from the original period, the more and more 
freakish becomes the weighting. This method has now been 
employed by the Economist in a series covering fifty years. It 
is amusing to think what queer weighting may be involved in 
comparing the fiftieth with the forty-ninth year. Yet, after all, 
this weighting may not be more incorrect than the even weight- 
ing; itself. 

That the comparison of the averages of prices using any prices, 
with apparent even weighting, may give an indefinite number 
of diiferent answers is, of course, simply due to the fact that an 
indefinite number of diiferent weightings may be used, according 
to the prices that happen to exist and to be hit upon at the 
first period. Yet this fact of the variability of the results ob- 
tainable has been urged as a reason for rejecting the system of 



190 MATHEMATICAL FORMULATION 

index-numbers altogether. Writers have toyed with various 
prices that might be used at the first period, and on the same 
price variations have naturally obtained different results. Then, 
not perceiviug the reason for the difference in the results, they 
have concluded that a method giving such a variety of answers 
can have no validity.^ Of course this criticism, based upon ig- 
norance, and merely playing off one incorrect form against an- 
other, has no validity against the correct form, whatever this 
may turn out to be when it is discovered. 

ISTow the variability of the results, it might be thought, could 
be obviated by confining our comparisons ahvays to the prices 
of the same mass-unit. If we did so with the intention of em- 
ploying even weighting, — in fact, employing even weightiug in 
the averages of prices at each period sej)arately, — there would 
still be uneven weightiug when we view the result as an aver- 
age of the price variations. For the prices would be large 
or small directly according to the preciousness of the classes, 
and so the result would be the same as if we averaged the 
variations using weighting directly accordmg to the precious- 
ness of the classes at the first period — an absurd system of 
weighting, as the importance of the classes is very different 
from their preciousness, and, in fact, apt to be the opposite. 
Moreover this method would not obviate the difficulty, as there 
would be two results obtainable according as we used through- 
out the same weight-unit or the same capacity-unit. Another 
objection, but of minor importance, is that this method could 
not possibly be applied to land, and with difficulty to gases. 
That it could not be apjjlied to labor, however, would be in its 
favor. We need not dwell longer on this method, since it has 
never been employed. But the notice of it is justified because 
its principal feature, the use of the same mass-unit, has been in- 
corporated in another method, to be examined presently. 

§ 4. Another way in which variability of the answers may be 

avoided is by taking account of the mass-quantities of the 

^ So Pierson in the paper referred to in a note to B. 122, and C. W. Oker, The 
fallacy of index-numbers, Journal of Political Economy, Chicago, Vol. IV., 1896, 
pp. 517-519. — The proper answer has recently been given by Padan, B. 141, pp. 
198-199. 



ERRORS INCURRED 191 

classes produced and consumed at one of the periods, or of some 
average mass-quantities, treating them as if they were the same 
at each period. This method has not infrecpiently been em- 
ployed, and may be called Scro])e'8 method, after the person who 
first suggested it (see Appendix C, IV.). Let x represent the 
number of the mass-units of the class [A] found to be produced 
or consumed by the community at large diu'ing a given period 
or periods, and y the number of the mass-units of the class [B] 
similarly found, and so on. Then at the first period the com- 
munity can buy this total mass of goods by expending the fol- 
lowing sum of money : xa^ -\- yi3^ -\- zy^ -(- to n terras, — the 

prices a^, |^^, y^, of course being the prices of the same mass- 
units of [A], [B], [C], whose numbers are represented by 

X, y, z, And at the second period the same total mass of 

goods can be purchased for this sum of money : xa.^ -f- yjS.^ -f- zy., 

-f- to n terms. If we divide each of these sums of money 

by the number of mass-units ])urchased — namely by x -\- y -j- z 

-f- to n terms, which we may represent by n' , — the quotient 

may be taken to be the average price of a mass-unit (an average 
mass-unit, so to speak) of the goods ; and therefore a variation 
of these averages may be represented thus, 

p -f ( -^'S + .'//^^ + s/'. + ) 

' ^(aj«i+;'/A + sri -h ) 

whicli reduces to 

P^ ^ xa^ + y/^2 + '^n + ^ 

■ Pi -^'"i + y^i + ^Ti + ' 

Or we might have started simply with this last formula as the 
expression of the \'ariation of the total prices of the same total 
quantity of goods. Therefore a meth<id doing this, although it 
does not necessarily involve an arithmetic averaging of the prices 
at each period, is the same as a method which does make such 
arithmetic averaging. 

There is little to recommend this method regarded as a com- 
parison of averages ; for the averages are average prices of a 



192 MATHEMATICAL FORMULATION 

rather nondescript mass-unit of goods. But these averages have 
the merit that, vary the mass-units Employed as we may, the 
two averages vary in the same proportion, so that the result of 
the comparison of them is unaffected ; wherefore also it is the 
same as that of a mere comparison of the total sums without any 
averaging at all. * Now viewed in this second form th re is 
much to recommend this method, provided the mass-quantities 
produced or consumed of every class are constant over both the 
periods compared. For then we are really comparing, so to 
speak, the total price at each period of the same mass of goods 
actually produced or consumed at each period. 

In this case we know, by our mathematical analysis, ^ that 
while in each of the separate averages of the prices the weights 

of the classes are x, y, z, , that is, the numbers of times the 

mass-units chosen are produced or consumed, yet in this method 
viewed as a method of arithmetically averaging the price varia- 
tions the weighting is according to xa^, yjS^, z)\, , that is, 

according to the total exchange- values of the classes at the first 
period, or, more simply, it is what we have called the weight- 
iug of the first period. This Ave have already denounced in the 
preceding Chapter as an absurd system of weighting. We shall, 
however, also find that this method may be viewed as a method 
using other kinds of averaging of the price variations, with other 
weighting, and so avoids the absurdity of using the weighting 
only of the first period. But we must postpone further exami- 
nation of this method, applied to this case, till a later Chapter. 
Here what needs to be noticed, is that in Scrope's method, in all 
its forms, regarded as a method of arithmetically averaging price 
variations, the weighting is according to xa^, y^^, zy^, 

* It is plain that all the terms used — yfi-^^, y(i2i ^7i> etc. — are the same whole 
quantities whatever be the sizes of the mass-units chosen. Cf. above Sect. V., 
Note 5. Hence it is indiiferent what mass-units be employed. This is shown at 
length by Padan, B. 141, pp. 195-198. Yet some writers employing this method 
have thought it necessary to state that we ought always to employ the same mass- 
unit. So Laspeyres, B. 26, p. 306, and Paasche, B. 33, p. 171. (Also Lehr has 
made a similar mistake with regard to his method, as will be pointed out on a 
later occasion.) In this method the variability of answers has already been 
avoided by use of the total money-values and elimination of the bare mass- 
quantities, and so does not need addition of the other method of avoiding varia- 
bility by the use of a single mass-unit. 

■''See Appendix A, II. ? 7 already referred to. 



KRHOliS I.NcrJtKEl) 1 {).'*> 

If the mass-quantities of any of the classes vary from the one 
period to the other, provided they do not all vary alike, tlie 
above recommendation for this method vanishes. For this case 
several variations upon this method have been suggested. Some 
writers have recommended using only the mass-quantities of the 
first period, others the mass-quantities only of the second period, 
and others a mean between these two, as ^ve have seen in the 
preceding Chapter. Furthermore, almost all the economists and 
statisticians who have advised the adoption of this method, or 
have actually adopted it, in any of its varieties, have had the 
habit, similar to the treatment of Carli's method, already noticed, 
of stringing out the com})arisons over many periods in a series, 
comparing each of the later periods directly with the first as a basic 
period. Some have used always the same mass-quantities, either 
those of the first period, or of some other one period, or a general 
average, in all these comparisons. Xow if the mass-quantities 
of the first or of some other one ])eriod be used, the comparisons 
between the later periods really are averages of the })r ice- vari- 
ations with weighting according to the money-values of the 
mass-quantities of the given period at the prices of the less 
removed of the two periods compared (or of the further removed, 
if the two periods be jirior to the basic period), — which is a 
wholly anomalous and senseless system of weighting, the weight- 
ing also being different in every comparison, though this was 
not intended. If a single general average mass-quantity of 
every class be used, the later comparisons are equally anomalous 
in form, but have the merit of not altogether disregarding the 
mass-quantities of the periods compared. There is little to 
recommend this method in theory, though it may perhaps turn 
out to yield ans^vers near to the truth in practice. If, lastly, 
the mass-quantities of the second, or later period, be used in all 
the comparisons with the first or basic period, a peculiar com- 
plication arises. For no^v, when we c(^mpare two later periods 
with each other, we are employing the mass-quantities of both 
these periods — that is, we ar(» employing double weighting, and 
double weighting ^vithout any authenticated method of selecting 
the mass-units, hence an altogether absurd method, struck upon 
13 



194 MATHEMATICAL FORMULATION 

by chance merely. This method^ without knowledge of its con- 
sequences, was actually recommended by Paasche, and may be 
called Paasche's variety of Scrope's method.'' The consequence 
which it brings about was evidently never intended ; for if it 
were, the same method of using the weighting of both periods 
ought to be employed in comparing the subsequent periods with 
the first, and if this double weighting is not wanted in those 
comparisons, it is not wanted in the others. In fact, Paasche 
himself objected to the use of double weighting, and criticised 
Drobisch for using it.'^ 

A similar inconsistency exists in another method. This is in 
the similar serial form of the method of arithmetically averag- 
ing the price variations with uneven weighting — the method 
first employed by Arthur Young. For if this method be em- 
ployed in such a series always with the weighting of the later 
period in every comparison of a later period with the common 
first or basic period, — as it has been employed by Mr. Palgrave, 
wherefore this may be called Palgrave's variety of Young's 
method, — then in the inverse of every comparison of prices be- 
tween any of the later periods (or in the direct comparison of the 
exchange- values of money) there is use of the weighting of each 
of the periods compared.® Here also the use of double weighting 
was never intended, and the same inconsistency exists ; for if 
the comparisons between the later periods ar.'^ correct, the com- 
parisons of the later periods with the first period cannot be cor- 
rect, and then correct comparisons would be dependent upon in- 
correct comparisons, — ^^vhich is absurd. 

§ 5. This unperceived merging of some of the common 
methods of employing single weighting into methods employing 
double weighting, suggests that we should here examine the 
methods avowedly employing double weighting (with the arith- 
metic averaging of prices), doing so in the comparison of the 
second period with the first as well as in every other comparison. 
But we may confine our attention for the present to the first of 
these methods ever invented, namely to Drobisch's, examining 

^ For its formula see Appendix C, IV. § 2 (2), and for the nature of its mass- 
tinits, Appendix C, V. § 4. 
■ B. 33, pp. 172-173. 
* For the formula see Appendix C, III. § 4. 



ET^noTlS IXCUKREl) 195 

thii^ because it also involves another of the features above no- 
ticed, namely the use of a single mass-unit in all the classes for 
the sake of avoiding; variability in the results. The mass-unit 
preferred by Drol)isch was a weiglit-iinit — a hundred-weight, 
though the size of the conmion weight is indifferent. He wanted 
the finding at each period of the average jiriee of tlie ^veight-unit 
of all goods, the average being the arithmetic with weighting 
according to the numbers of the weight-units in every class at 
each period. He claimed that by so doing we should get a sort 
of absolute average price, and so disprove Jevons's denial of an 
average price.** 

The formula for this method is the general fornuda for the 
arithmetic average of prices with double weighting above given 
(17, 2,3), the method of selecting the mass-unit being presup- 
posed. The method itself may be illustrated by the following 
example.'^ If in a country during a certain first period say ten 
millions of the same weight-units of goods (this figure being ob- 
tained by adding up all the numbers of weight-imits of all the 

classes — .i\ + //, -f- 2, -f- to n^ terms = 10) are produced and 

consumed at a total valuation of fifteen million money-units 
(this sum being obtained by adding up the numbers of weight- 
units of every class multiplied by their prices — x//.^ -\- y^^^ + zj^ 

-(- to /(j terms = lo), there is an average price, represented 

bv i| or \l money-units, for the weight-unit. Then if at the 

second period twelve million weight-units {x.^ + y^-^- ■■^2 + 

tt) n.^ terms =12) are produced and consumed at a valuation of 

twenty-one million money-units {x./i,^ -\- yjd.^ -\- z.-,'.^ -\- to n^, 

terms = 21), the average of prices, or the average price of the 
mass-unit, would be |i or If money-units. Therefore, he would 
conclude, not merely an average of prices, but simply the price 
of the mass-unit of goods, has risen from IJ to 1|, or as from 
6 to 7, that is, by 16f per cent., and inversely money has de- 
preciated as from 7 to 6, by 14.29 per cent. 

In this method the use, not of the arithmetic average, nor of 
double weighting, but of a common mass-unit may be sho^vn to 
be wrong. 

9B. 29, pp. 44-45, B. 30, p. 153. 
1" See also Appendix C, V. ^ 1. 



196 MATHEMATICAL FORMULATION 

The use of a common mass-unit means that the aA^erage drawn 
at each period is an average price of an average mass, so to 
speak, of all goods, or rather, it is simply the price of a small 
mass made up of small fractious of all goods, these fractions 
being proportioned to one another, by weight or by bulk, accord- 
ing as are the weights or the bulks of the classes relatively to 
one another. This small compound mass, or Anaxagorian 
homoeomeria, is a sort of representative mass- unit, being com- 
posed in the same way as the whole mass of goods. It may be 
diiferently composed at each period, and its price may be differ- 
ent at each period. Drobiseh claimed that the variation of its 
price represents the variation in the money-value of all goods, 
and inversely the variation of the exchange-value of money in 
all goods (other than money). This claim is unfounded, and 
violates several principles. 

In the first place no reason is offered why the use of a com- 
mon weight-unit was chosen rather than a common capacity- 
unit. Yet two differently varying prices would be obtained 
according as the one or the other is used of these kinds of mass- 
units. Thus this method does not succeed, except arbitrarily, 
even in the attempt to a^^oid the fault of admitting variability 
of equally good and hence mutually destructive results, and suc- 
cumbs to a criticism already advanced against several other 
suggested or possible methods. 

In the second place, because of this feature in it, the follow- 
ing absurdities can be proved of this method : (1) between two 
periods between which no variation of any price whatsoever 
takes place, if any irregular change takes place in the mass- 
quantities, the result wdll indicate a variation of prices " (con- 
trary to Propositions XXVII. and XLIY.) ; (2) between two 
periods between which all prices have varied in exactly the same 
proportion, if any irregular change takes place in the mass- 
quantities, the result will indicate a variation different from 
that common variation (contrary to Propositions XVII. and 
XLV.) ; (3) between two periods between which all prices have 

11 This is the criticism made by Laspeyres, B. 26, p. 308. Drobiseh replied by 
I'eaffirming his position, asseverating that tliis is as it should be, B. 31, pp. 425- 
42(3. 



KlUJOIiS [XCrilRED 197 

risen somewhat, it" certain changes take place in the mass-quan- 
tities, it is possible that the result may even indicate a fall of the 
a\'erat>e price ; and conversely/^ These absurdities are not 
due to the use of double weighting, but to the use of double 
weighting along with this method of selecting a common mass- 
unit. They are to be found in some other methods using double 
M'eighting, where they are due to its conjunction with some other 
(conse(|uently) improper method of selecting the mass-units. 
Thus all three are found, upon inspection of the formulae, in 
Paasche's variety of Scro})e's method and in Palgrave's variety 
of Young's method, so far as these use double weighting, that is, 
in comparisons between later periods, in a series founded on one 
period as a base. In Lehr's method, rather curiously, the second 
and third absurdities are to be found, but not the first. Again 
all three arc; foimd in the last two forms into which a rather 
vague method invented by Professor Nicholson may be analyzed. ^^ 
But a method using double weighting with the arithmetic average 
may be discovered that is free from all these absurdities. 

In Drobisch's method it is not difficult to see why double 
weighting with use of a common mass-unit leads to these ab- 
surdities. All prices remainmg constant, an increase m the 
quantity of expensive goods, the prices of which are above the 
average at the first period, tends to raise the average at the 
second period (and reversely a decrease) ; but an increase in 
the quantity of cheap articles, the prices of which are below the 
average at the first period, tends to lower the average at the 
second (and reversel}' a decrease) ; and the average will be 
changed according as the one or the other of these movements 

^'^ E. g., in the abovo mimerii-al example, the following variations would be 
possible. Suppose at the tirst period things were thus : 

cheap goods 8 mill, cwts., value 8 mill, dollars ; price | = 1.00, 
costly " 2 " " " 7 " " " i-3.50, 

total 10 " " " 15 " " " H = l-50; 

and at the second thus : 

cheap goods 7 " " " 6 " " " f = .86, 

costly " 5 " " " l."i " " " i!i = 3.00, 

total 12 " " " 21 " " " ti=l-75. 

Here the whole collection of cheap goods has fallen in price, and also the whole 
collection of costly goods, wherefore it is possible that every article has fallen in 
price. Yet the average of the whole has risen in price. 
1* See Appendix (', V. '!> :>. 



198 MATHEMATICAL FORMrLATION 

predominates. And all prices rising, or their average variation 
being a rise, the former change of quantities will raise the aver- 
age still higher, and the latter will prevent it from rising so high 
— -perhaps will leave it at constancy, or even occasion a fall ; and 
reversely if all prices fall, or their average variation is a fall/* 
It is evident that the disappearance of a cheap article and the 
appearance of a dear one have the same eifect as the variation 
of a cheap article into a dear one ; and reversely the disappear- 
ance of a dear article and the appearance of a cheap one. A 
single disappearance, or a single appearance, has half this influ- 
ence. Thus, prices remainmg constant, or varying in a given 
degree, whether this method shall indicate a rise or a fall of 
prices, or a greater or a smaller rise or fall, will depend upon 
whether the change in the weighting happens to fall more on 
expensive goods or more on cheap goods. ^^ 

Thirdly, even at each of the periods separately the weighting 
employed by Drobisch is wrong ; for his weighting is according 
to the mass-quantities, which weighting we have seen to be im- 
proper (in Chapter lY., Sect. III. § 3). A reason why it is 
incorrect is that the mass-quantities used have diflPerent precious- 
ness, which also deserves to be counted, or otherwise the height 
of the average will depend upon the accident whether dear or 
cheap goods happen to abound. Still, if we tried to correct 
this error by weighting the mass-units at each period according 
to their preciousness, as well as according to their masses, we 
should get no determinate result, because at each period the 
average would return to the lowest price we happened to take 
as our basis (probably a mass-unit of the poorest quality). The 
two averages would have no connection with each other, and 
the comparison of them would have no meaning. To get this 
connection we need to employ a mass-unit in every class that 

1* Thus in the example in Note 12 the supposed increase of the falling goods 
was in the expensive goods and the decrease in the cheap goods, and the result 
was a rise. — The general mathematical principles are given in Appendix A, YII. 
g 4, with reference back to I. §§ 8-10. 

15 Hence Drobisch's method can be true only if the mass-quantities do not 
vary, or all vary in the'someproportion. This was perceived by Lehr, who like- 
wise condemns Drobisch's method for using such economically dissimilar mix- 
tures of things, B. 68, pp. 41-42. The last criticism is also made by Zuckerkandl, 
B. 115, p. 247, B. 116, p. 24o 



P 



A/^^ 



EKItOHS INcrRRP:!) 199 

has the same exchange-vahie over l)<)th the })eriods together as 
in every other class. This is the iinprovement introduced by 
Professor Lehr, whose method, however, as we shall find in a 
later Cha})ter, does not correctly carry out the improvement in- 
tended, wherefore the true method is still something else. 

Lastly, a general reason why Drobisch's method is wrong is 
that it lias been invented without any regard for the principles 
of simple mensuration. We have seen in Chapter III. that we 
must compare variations only in two similar worlds. Neither 
of the methods there pointed out for obtaining such similarity 
are here observed. The same trouble exists in the varieties of 
Scrope's and of Young's methods introduced by Paasche and 
by Palgrave. In comparing a later period with a first period 
on a given \veigliting, we are comparing variations in two sim- 
ilar worlds, whicli so far is correct ; and in comparing another 
later period with the first on another weighting, we are eompai'- 
ing variations in another set of similar worlds ; but in compar- 
ing the results of tJiese two measurements we are comparing 
things in two difierent worlds. 

It should be noticed that in many — in fact, in most — subjects 
of statistics we do wish to use double weighting Avith the quan- 
tities as they happen actually to l)e reported, that is, to com- 
pare averages at different periods built on varying quantities of 
thino:s whose indivndualitv is reckoned with(^)ut regard to the 
attribute we are measuring. Here the above principles of 
simple mensuration are not violated, for the good reason that 
these are not subjects of simple mensuration. And if we ai'e 
comparing things in dissimilar worlds, the very point is tliat 
we are dealing with dissimilar Avorlds and want to compare 
them. Thus to measure the measure (^f length at different 
periods, Ave must deal with similar worlds at both periods ; but 
in measuring the average tallness of people at difierent times 
or jjlaces, we are expressly dealing with difierent peoples, and 
so must average their heights at each period or place sej^a- 
rately, weighting each operation according to the numbers of 
persons measured, and then merely compare the results of the 
separate averagings — in all eases using a supposedly constant 



200 MATHEMATICAL FOEMULATIOX 

measure of length already examined and approved. Again, to 
use an example nearly allied to our own subject, in the measure- 
ment of the average wealth of a country at diiferent periods we 
must take into account the different total wealths and the differ- 
ent numbers of the populations possessing them. Here, for in- 
stance, it would be perfectly proper for the result to indicate 
that, though there be no variation whatsoever in the wealth of 
the individuals in the diiferent classes of society (as found by 
our using a measure of wealth already examined and approved), 
yet if the wealthier classes increased more in numbers, the aver- 
age wealth of the country \vould increase, and if the poorer 
classes increased more in numbers, the average wealth of a 
country would decrease. 

Not so, however, in regard to the exchange-value of money 
(which is the measure itself used in the preceding measurement). 
Suppose a creditor should argue thus : " Prices have not varied 
a particle, but I, and the whole community as well, are now pur- 
chasing more expensive articles and fewer cheap articles than we 
used to do ; therefore, as we get a smaller quantity of goods for the 
same money, our money has fallen in purchasing power or ex- 
change-value, and the average of prices has risen, and my debtor 
ought to pay me back more money than he borrowed, to make 
up the deficiency." It is plain that he would be talking non- 
sense, because the debtor could readily reply : "If you and I 
and the rest of us spend our money more on expensive articles 
and less on cheap articles than we used to, the fact that prices 
have not changed shows that, though we are getting smaller 
quantities, we are getting better qualities, and the gain in the 
latter makes up for the loss in the former •^'' wherefore there is 
no reason why I should give you back more money to enable 
you to get the same quantity of better articles." 

Now if we already had a correct measure of exchange-value, 
if we wanted to measure variations of the average preciousness, 
not of money, but of all commodities in general, we should jjursue 
exactly the course advised by Drobisch for measuring variations 
of the average exchange- value of mouey. It is perfectly proper 

>« Cf. Chapt. IV., Sect. V., Xote li>. 



ERIIOIiS INcrfiREI) 201 

that, the preciousness of individual thin<i;s remaininji; unchanged 
(all prices being constant), an increase in the (piantity of cheap 
things should lower the average of })reciousness, and an increase 
in the quantity of costly things should raise it. The three ab- 
surdities in Drobisch's method as a measure of the exchange- 
value of money cease to exist in it as a measure of the precious- 
ness of commodities. We have seen, also, that Drobisch's 
method gives two results, according as it is applied to a commim 
weight-unit or to a common capacity-unit. This is perfectly 
proper in the method as a method of measuring the ])reciousness 
of goods, because there are two ways of conceiving of precious- 
uess.'" Thus Drobisch has hit one thing while aiming at another. 
But the pro^•iso stated at the commencement is of importance. 
For the perfect working of this method of measuring variations 
in the preciousness of goods the exchange-value of money, by 
which we measure the preciousness of individual things, ought 
to be constant, and we should know this. '^ Otherwise its varia- 
tions must be allowed for; for which purpose also its variations 
need to be known. Thus the proper method of measuring 
variations in the exchange- value of money is a prerequisite even 
for the employment of Drobisch's method as a method for meas- 
uring variations in the preciousness of all commodities. 

§ 6. Geometric averaging of prices. — As in the case of aver- 
aging the exchange-values of money, the geometric average of 
prices does not fall into two different methods according as we 
compare averages of prices or average the variations of prices. 
^^^hat is an apparent weighting in the one is actually the weight- 
ing both in it and in the other. If we average prices at two 
periods separately by using weighting, the same in both cases, 
according to the relative mass-quantities, the same result is ob- 
tained as if we employed the same weighting in averaging the 
variations of prices. But we know that we do not Avant to 
weight either the prices or the variations according to the mass- 
quantities — or the mere numbers of times some mass-units hap- 
pen to recur, — but according to the total exchange-values, which 

^" See above Sect. III., Note 3. 

^^ Money serves here as water does for lueasiiriiig the specific gravity of boilies. 



202 MATHEMATICAL FORMULATION 

correspond to the numbers of times equivalent mass-units are 
repeated. Hence the average should be in this form, 

P„ 






p' = l/ I -') (^ I •( - ) • tonterms, 



in which n" = a + b -(- C -(- to n terms, and a = xa, b = yj^, 

and so on. At which periods the terms xa, y^, should be 

chosen, we have examined in the preceding Chapter, and have 
concluded that the best weighting is for a to be equal to Vx-^a^x^a^, 
b to 1^2/1/^13/2/^2? ^^^ ^^ ^^- ^^® reason why weighting accord- 
ing to the mere mass-quantities is not to be employed is because 
this weighting would be accidental according to the sizes of the 
mass-units used. Or if, to avoid this variability in the results, 
we used the same mass-unit throughout, there would still be a 
choice between two results, on a unit of weight or a unit of ca- 
pacity. And in either of these cases, as, given the total ex- 
change-values of two classes to be the same, the same mass-unit 
would be repeated less frequently in a more precious article than 
in a less precious one, the weighting would depart from the proper 
weighting inversely according to the preciousness of the articles. 
§ 7. In a series of successive periods, if we employ even 
weightmg or the same uneven weighting throughout, a compar- 
ison of subsequent periods all with a first basic period, on the 
geometric average, will yield results which, compared with one 
another, give the same results as when these periods are directly 
compared Avith one another with the same weighting. Thus the 
comparison between the second and the first will be as above 
indicated, that between the third and the first like unto it with 
change of numbering, and that between the results of these 
comparisons as follows, 

which reduces to 



ir \i) W \n) '""*^'™^' 






V 



KJIRORS IXCURRED 20.'> 

which is the formula for comjjaring the third period with tlie 
second period directly. 

This agreement does not exist in the comparisons made in 
series constructed on Carli's or on Young's method, or in a few 
other ways, as in one variety of Scrope's method. The employ- 
ment of these methods on the usual plan of comparing all later 
periods directly with an original basic period involves an incon- 
sistency, since the indirect ct)mparisons between these later 
periods yield a different result from what would have been 
yielded by the employment of the same method of averaging in 
a direct measurement between these periods, or from what the 
indirect measurement itself would have been had a different basic 
period been chosen.'^ This inconsistency appears never to 
have been perceived ^° until it was pointed out a few years 
ago by Professor Westergaard. Professor AVestergaard also as- 
serted, in a general way, that this inconsistency does not occur 
in the geometric method, and advanced this difference as an ar- 
gument in favor of the geometric method.-' Credit is due to 
Professor Westergaard for calling attention to this difference. 
But his use of it as an argument for the geometric averaging ( if 
price variations suggests the following remarks. 

First, there are other methods beside the geometric that share 
with it freedom from this inconsistency. Thus in a series in 
which the later periods are separately compared with the first, 
this inconsistency is not incurred by the use of Dutot's metluKl, 
V)y the use of Scrope's method with the same mass-(piantities in 
all cases (either those of the first or of some other jieriod, ov a 
general average of all the periods or of some of them or of any 
periods),^^ or by the use of Drobisch's method (provided it be 
confined to the same classes). Hence the argument for the 
geometric average as exclusively possessing a certain excellence, 
is wholly invalid. Professor Westergaard had in mind only 

18 See above in ? 3 in regard to Carli's method. It is plain that the same liolds 
good in Young's. Cf. Appendix C, II. § '2 and III. § 2. 

~° A good example of how it has been ignored is furnished by De Foville. See 
Appendix C, IV. § 2 (2). 

21 B. 110, pp. 219-220, followed by Edgeworth, B. 65, p. 380, B. (30, p. 137. 

-- See the formulae for a series of index-numbers using this metliod, in Appi n- 
dix C, IV. 'i 1. 



204 MATHEMATICAL FOEMULATIOX 

Carli's and Yoimg's methods ; and his argument is due to this 
restriction. As for the methods which do incur this inconsis- 
tency, these are not only Carli's and Young's, including Pal- 
grave's variety of the latter, but also Paasche's variety of 
Scrope's, Lehr's, Nicholson's (in its one original form), and an- 
other which is still to be described — in the last three provided 
they were to be employed in the serial form with dependence 
upon a common base.^^ 

Secondly, it should be distinctly noticed that even the geo- 
metric average possesses this excellence only if used with the 
same weighting through the whole series. But this system of 
weighting evidently is slipshod, and cannot pretend to give 
perfectly true results. The true system of weighting is to em- 
ploy special weighting in every single comparison, according to 
the conditions existing at the two periods compared. Now if 
^ve employed such proper weighting in comparing the second 
with the first, and again in comparing the third with the first, 
the Aveighting would most likely be different in these two 
comparisons. Then a comparison of the results so obtamed 
A\ould, as in some of the arithmetic averagings already noticed, 
be a comjjarison between these two periods with double weight- 
ing — and of such a kind as to lead to the three absurdities into 
^vhich Drobisch's method fell. And now the direct comparison 
between the two periods, with single weighting, would be witk 
weighting still difPerent from that used in either of the preced- 
ing comparisons, and its result would not agree, except acciden- 

-^ In all these methods the direct comparisons between the later periods, and 
the indirect comparisons between them through mediation of their comparisons 
with the basic period, may agree under certain conditions. It is easy to discover 
these conditions by comparing the formula, and it may be interesting to state 
them. Thus tlie indirect agrees with the direct comparison in Carli's method if 
between the basic and the nearest subsequent period (or furthest prior period) 
there be uo price variations, or if all these be in the same proportion ; in Young's 
method, the same ; in Palgrave's variety of Young's method, if the mass-quantities 
be the same at the two subsequent periods, or if they have varied between these 
two periods in the same proportion ; in Paasche's variety of Scrope's method, if 
between the two subsequent periods there be no variations either in the prices or 
in the mass-quantities, or if between these two periods both the prices and the 
mass-quantities have all varied in one and the same proportion. The other 
methods have never been suggested for use in this serial form. What would be 
the conditions of such agreement in them if so used will be examined for another 
purpose in a later Chapter. 



ERRORS INCURRED '205 

tally, with the result obtained by eonn)aring' their results. Yet, 
supposing the geometric average the right one to use, among 
these comparisons it is only this direct one, if the jieriods be 
contiguous, that can give the correct result. 

Thirdly, it would seem to be unquestionable that the agree- 
ment between the indirect and the direct com})aris()ns ought to 
exist. Professor Westergaard based his recommendation of the 
geometric average upon the principle that prices measured from 
1860 to 1870 and from 1870 to 1880 ought to show the same 
variation from 18 GO to 1880 as would be shown by comparing 
the prices of ISSO directly with those of 1800."^ By adding 
another supposition this can be made self-evident. Suppose 
that everything at the last period is in exactly the same state as 
at the first (or that everything diverges in the same ])roportion). 
Then it is evident that the exchange- value of money is the same 
at the last period as at the first period (or deviates inversely to 
a uniform ])rice variation), whatever intervening variations may 
have taken place.'^ Constancy (or that variation) is sho\vn l)y 
the direct comparison between these two periods ; and constancy 
(or that variation) should be sho^v^n by a series of comparisons 
through intervening periods, no matter how many or how varied 
the intervening periods may be. But it does not follow that this 
agreement should be sought at the expense of other require- 
ments, or that such agreement, if existing in any method, can 
prove it to be true, if we already know that this method fails in 
other respects. For instance, Drobisch's method provides this 
agreement, but Drobisch's method is not thereby proved to be 
correct. The need of this agreement can be taken only as a 
negative test, or touchstone. If a method has other things to 
recommend it, it may be dispnjved by being found not to stand 
this test. But unless it is recommended by other reasons, it is 
not proved by its satisfying this test. Perhaps, however, the 

24 B. 110, p. 219. 

25Thiswould seem to be so also if ouly all the priceswere the same (or equally 
divergent) at the separate periods, let the mass-quantities be what they may — in 
accordance with Propositions XXVII. and XVII., and XLIV. and XLV. But 
if the mass-quantities as well as the prices are all exactly the same (or equally 
divergent), there can be no question about it. 



206 MATHEMATICAL FORMULATION 

perfectly correct method^ for a long sequence of periods, will not 
be found to be workable, and the workable form that comes the 
nearest to it may not satisfy this requirement perfectly. Then 
this requirement may be used as a means of correcting the latter. 

Lastly, this argument of Professor Westergaard's leads to the 
consideration whether ^^^e should continue in the usual course of 
comparing every subsequent period with the same original 
period as common basis for a whole series, or whether the series 
should be formed by comparing every period with the immedi- 
ately preceding period, and again the next period with it, and 
welding them all together. The latter is as practicable a course 
as the former, and may yield the same sort of serial index-num- 
bers. For example, if we find the result of the first comparison 
to show that between the first and the second periods prices have 
fallen on the average by 10 per cent., and the result of the 
second comparison to show that between the second and the 
third periods they have risen by 30 per cent., we may express 
these relations in the series 100, 90, 120, which indicates at 
once that, by passing through the second period, the level of 
j)rices at the third is 20 per cent, above what it was at the first. 
As to the first of these procedures, there is little in its favor but 
its convenience. The most natural and rational procedure, the 
moment we attach importance to correct weighting, would seem 
to be the second. At first sight the argument between the two 
appears to lie merely between the opinions that in the former we 
gain exactness in the comparisons of the later periods with the 
first and lose it in the comparisons between the later periods, 
and in the second we gain exactness in the comparisons between 
contiguous periods and lose it in all other comparisons. But of 
course both these positions are false, being self-contradictory. 
The question is really. In which of the two courses are we likely 
to gain greater exactness in the comparisons actually made? 
Here the probability seems to inclme in favor of the second 
course ; for the conditions are likely to be less diverse between 
two contiguous periods than between two periods say fifty years 
apart. 

§ 8. There is still another reason for adopting the second 



ERRORS INCURRED 207 

course. This is that the first course is tied to the employraent 
of a definite number of classes, decided once for all at the start ; 
but the second course permits the dropping at any time of an 
old class or the introduction of a new one. In fact this second 
course is the only one permitting an accurate measurement of the 
exchange-value of money over a very long series — say of one or 
several centuries. For in periods separated by a very long in- 
terval there is much that is not common to both the periods, and 
to compare them by what is common to them both is to make a 
restricted and incom])lete comi)arison. A complete comparison 
can be made onl}' by comparing all that is in each of them with 
the intervening periods near to them, gradually dropping out the 
old things as they disappear and introducing the new as they 
appear. It is also only in the comparison of such temporally 
distant periods that the divergence between the two ways of 
conducting the measurement is likely to be great. Therefore 
for long series it is important to decide betw^een the two courses. 
Now if the two periods be two years separated by a century, it 
is probable that if a rightly conducted measurement had been 
made of the variations from year to year filling up the whole 
century, the result at the end would be very nearly the true one, 
while a direct comparison between the two extreme years, which 
comparison might have t(j leave out altogether some classes that 
have been counted in the other comparisons, because they did 
not exist at the first period or no longer exist at the last, might 
be appreciably divergent from the truth, even though the com- 
parison were made in proper manner and the data were correct.^^ 
Hence the procedure of comparing every successive period with 
the immediately preceding is the proper one to adoi)t if it be 
intended to continue the measurement over many years, or in- 
definitely."" 

Incidentally ^ve hereb}- see also that among the reasons why 
we cannot successfully compare the exchange-value of money 

2s Cf. above Chapt. IV., Sect. V. jJ 7, Note IS. 

2 7 Into this course De Foville was forced by eirc\iiustanees in his calculations 
after 1862. It has been recommended by Lehr for his method, B. 68, pp. 45, 46, 
and in general by the British Association Committee, Firfit Report, B. 90, p. 250, 
and Edgeworth, B. 59, pp. 268-269. 



208 MATHEMATICAL FORMULATION 

to-day with Avhat it was five hundred years ago is not merely 
that we have not accurate data about the year 1400, but that we 
have not complete data about all the intervening years. Yet 
there is no reason why, if from now on the measurement be cor- 
rectly made from year to year, the people living in 2400 should 
not know with almost perfect exactness the relation between the 
exchange-value of their money and ours.^^ 

The brief hypothetical series of index-numbers above used as 
examples, namely 100, 90, 120, can be reduced to this : 111.11, 
100, 133.33, or again to this : 83.33, 75, 100, or to any other 
series in which the figures observe the same proportions.^^ This 
fact shows that it is of no importance what period is selected as 
the so-called basic period of the series, whichever of the above 
two courses be adopted. Properly every period is basic in the 
comparison with the next period, and none is basic for a series 
of periods. A series with the index-number 100 at one period 
does not better represent the sequence of events than a series 
with this index-number at any other period. ISTor is there 
rhyme or reason for making the basic period longer than the 
other periods. Much discussion as to the superiority of this or 
that base in the various attempts at measuring the course of the 
exchange-value of money during the latter half of the century 
just elapsed has been wasted.^'^ 

YII. 

§ 1. There remains only to point out the mathematical formu- 
lation for the exchange-value of money in all things (or inversely 
for the level of all prices, including the price of money). This 

2* The not infrequent complaint about the incommensurability of the exchange- 
value of money at two long separated periods directly compared (e. g., recently 
by W. Cunningham, On the value of money, Quarterly Journal of Economics, 
July 1899, pp. 379-385) should be allowed no influence in the question as to 
whether the variation of this exchange- value through successive periods is meas- 
urable. The difficulty is greater in the attempt to measui'e the contemporaneous 
levels of the exchange-value of money in two distinct countries, say an arctic and 
a tropical. But luckily we have not so mucji interest in measuring this. 

2^ The table of Evelyn on the basis of 1550 was thus turned about and put on 
the basis of 1700 by J. P. Smith, B. 7, pp. 472-476. 

30 Concern for the base was first shown by Porter, B. 11, p. 440. That the base 
should be carefully selected is the first of the canons laid down by Martin, op. cit., 
p. 626. It is said to be of great importance by Mayo-Smith, B. 137, p. 486. 



OF EXCiiA\(;i:-VALri-; in all tiiin(;s 201) 

is easy on the assumption that any of the mathematical averages 
is proper for expressing tlie exehang-e-vahie of money in all otlwr 
things. We simply have to add to the right side of the equa- 
tion the expression for the exchange-value of the money-unit in 
money, which is always unity, and invariable. 

Thus, taking the arithmetic average and using M„ to exjiress 
the exchange-value of the money-unit in <dl things, we have for 
the first period, with even weighting, 

3/ai= ^-^-^-j {1 + («^) + i^B) + (eC) + to n + 1 terms}, 

which reduces to unity. And for the second period, again with 
the same even weighting, we have 

il/a.= ^^{1 + a\aA) + h'{bB) + c'{cC) 

-\- to » + 1 terms}. 

And the comparison of these two reduces to 

M 1 

l£ = ,T^:l(l + "' + 6' + C + to n + 1 terms). (IS, i) 

For the other kinds of averaging, still with even weighting, 
we should have, after similar reductions, 

Ma2 _ 1 

n -\- 
and 



1/111 \' 



Ma^^ ^ "j^^a' b' c' ■ to n terms. (18, 3) 

It is easy to restate these formulfe with the terms — , -r, —, 

to represent the variations, in place of a' , b', c', 

Inversely, by means of averages of prices, the same results 
mav be obtained bv inserting in the formulae for the averages of 
price variations the price also of the money-unit, which is in- 
variably a unit. 

With uneven single weighting, we have to add the weiii^hts of 
14 



210 MATHEMATICAL FORMULATION 

the commodity classes as before, and in addition we must give 
some weight to the class money. What the nature of this weight 
is, will be left over for a later discussion (in Chapter XIII.). 
Here we may represent it by m. Then, using the form for the 
variations of prices, with P„ to represent the level of prices with 
the price of money included, we have 

P«i 1 



^«2 1 /_ I „ «i I T, /^i I „ ri 



(19, 2) 



^/■/_L«, (m + a^ + b ^ +cJ-^+ ton + l terms) 

S?=.-^Tin(«^ + <t + ^|+'n+ ton+lternas), (19,2) 



Pa 2 ""t"! 
Poi' 



It is evident on inspection that, compared with the results of 
the formulae for the corresponding averages of the variations of 
money in exchange-value in all other things, the results of these 
formulae always show a smaller variation than those, in agree- 
ment with Proposition XXI. ; that they aRvays mdicate con- 
stancy when those do, in agreement with Propositions XXII. 
and XXVI. ; that they always vary in the same direction, in 
agreement with Proposition XXIII. ; that their divergence, or 
falling short, from those is always smaller the greater the num- 
ber of terms (or rather, the larger is n" compared with m), in 
agreement with Proposition XXIV. ; and that the proportion 
of this divergence is always the same, in agreement with Propo- 
sition XXV. ; and lastly that their indications are unaifected by 
the numbers of other things that are constant in general ex- 
change-value, if the indications be of constancy, or that vary 
in the same proportion in exchange-value in all things (their 

prices varying as p^ J, in agreement with Propositions XXXII., 

XXXV. and XXXVI. 

§ 2. Other formulae may be altered in the same manner. In 
Scrope's method the want of body to money — the indiiference 
we all feel to its mass — causes no trouble, because all the terms 
are the total money-values of the classes, and so long as we can 
get this in the case of money, without reference to its weight or 



Ol' KXCHANCiH-VAHi: IX ALL TIlL\(i.S 211 

bulk,' we :uv satisfied. Thus, with v representini^ the number 
of money-units employed, the formulse for Scrope's method are 

tlic fdlldwinii', 

K>^ '■ + ''"S + M + "-/'2+ 

P„, r + .*•«, + ,y/9, 4- 2/'] + ' 

and 

Ma, ^ r + -m, + //;9, + zr, + 

M„, V + xa^ + i/ft^ + zj'2 + ■ 

The same remarks a})]>ly to these as have just been made on the 
formuhe for averaging the })rice variations (except the reference 
to Proposition XXV.). - 

In Drobiseh's method, however, where specific weights, or 
capacities, appear alone in one half of the formula, it would be 
impossible to make the alteration permitting this method to be 
applied to the measurement of the exchange-value of money in 
all thintrs. Here is one more reason showing: the falsitv of that 
method. - 

But there are methods employing double weighting that are 
applicable to the measurement of the exchange-value of money 
in all things, including itself. 

1 The mass-unit of money which we habitually use is nothing else than the 
money-unit itself, whose money-value is always one money-unit. 

- As a measure of the preciousness of commodities it is no fault in this method 
that it cannot be applied to all things, including money. For money, in which 
mass is inessential, has no preciousness. 



CHAPTER Aa. 

THE QUESTION OF THE MEANS AND AVERAGES. 

I. 

§ 1 . We are now prepared for entering upon our subject proper^ 
the search after the right method of measurmg general exchange- 
value and its variations. We must deal first exhaustively with 
exchange-value in all other things, as the simpler. 

When one thing rises or falls to a certain extent in exchange- 
value in every other thing alike, we know (by Proposition XVII.) 
that it rises or falls to the same extent in exchange-value in all 
other things, and (by Proposition XVIII.) that every one of the 
other things have reversely fallen or risen to a smaller extent in 
exchange-value in all other things. We also know (by Prop- 
osition XIX.) that the larger the number of the things, the 
smaller is the opposite variation of every one of the other things. 
But we do not as yet know how great the opposite variation of 
every one of the other things is, given the variation of one and 
their number. Here is a Proposition lacking, which if we shall 
be able to supply, we shall be able to measure all exchange-value 
variations. The problem, then, is to supply this Proposition. 

In attempting to solve a problem it is well to put it in its 
simplest form. The simplest form in Avhich we can put our 
problem is the following. Let us suppose that we are dealing, 
with a world in which are only three classes of exchangeable 
things (or with a part of our world which consists of only three 
classes). One of them let us suppose to be money, so that we 
have a world with money, represented by [M] , and two classes 
of commodities, represented by [A] and [B] . Here all the 
other things beside the one class, money, whose exchange- value 
we are examining, are comprised in only two classes. These 

212 



Tin-: IM;()l',I,K>t AND SUGGESTED SOU'TFONS 213 

two classes of commoditios we may also, for simplicity, suppose 
to he equally important, or equally larji^e — not at one period only, 
hut either at each of the periods compared or somehow over hoth 
the periods together, without reopening here the question how 
their importance or economic size is to be computed. Now if 
we succeed in solving the problem in this confined and simple 
form, Ave shall not be able immediately to extend the same solu- 
tion to all cases; for what is true of two equally important 
classes may perhaps not admit of being applied to more com- 
])lex cases. But if we succeed in the simplest case, we shall be 
AN ell on the way to solving the complex cases. These, more- 
over, may be admitted to consideration as we advance further, 
even before we finally solve the simple cases. 

Simplicity in the fi)rm of the problem will also be carried out 
by supposing that at tlie first period the money-unit is equiva- 
lent to a mass-unit of each of the commodities, so that M=c=A-c>B, 
and the prices of A and B are one money-unit each.' We may 
then suppose that at the second period A rises in exchange-value 
in the other two, individually and together, by one half, or by 
•')() per cent. ; which means that its price rises to 1.50, while the 
price of B remains 1.00. At the same time M and B have 
each fallen by one third in exchange-value in [A], or by 33 J 
per cent., but as M retains its exchange-value in [B] and B its 
in [M], M has not fallen as much in [A] and [B] together, 
that is, in exchange-value in all other things, nor has B fallen 
so much in [A] and [M] together, that is, also, in exchange- 
value in all other things. 

Digressively it may here be remarked that, as a corollary to 
Proposition XIV. [M] and [B], that is, tico or more classes 
that retain the same exchange-value in each other, vary alike in 
exchange-value in all other things beside themselves and in exchange- 
value in all things including themselves, hut not in all other things 
(i. e., other to each class), unless these cla^sses be equally large 
{over both the periods together), (Proposition XLVI.). The 

^ But we must be ou our guard not to regard A and B, equivalent at the first 
period only, as the ecououiie individuals in the classes [A] and [B] for the two 
periods compared ; for the economie individuals must be equivalent over both 
the periods together. 



214 THE QUESTION OF THE MEANS AND AVEEA(iES 

reason why they do not necessarily vary (or remain constant) 
alike in exchange-value in all other things, is because for each 
the standard of other things, composed of the classes other to it, 
is diflPerent from what it is for every other.^ Thus, in the ex- 
ample before us, if [M] be a class twice as large as [B], and 
consequently of [ A] , the rise of A will have less influence in 
depressing the exchange-value in all other things of [B] than in 
depressing the exchange- value in all other things of [A] , since 
it is a rise of only one third of the things other to the class [B] , 
while it is a rise of one half of the things other to the class [M] . 
But if the classes [M] and [B] are equally large or important, 
the class [A] will be of the same relative size among the things 
other to each of them, and hence will have the same influence in 
making them vary in exchange-value in all other things. We 
do not here concern ourselves with the size of the class [M], 
and confine our attention to enquiring what influence upon its 
exchange-value in all other things has a rise of the exchange- 
value of [A] both in it and in [B] . For this purpose we must 
presuppose knowledge of the relative sizes of [A] and [B] ; 
and, to repeat, the simplest case is to suppose that they are equally 
large. 

Our special problem then is : Given the above supposed con- 
ditions and variations of A, how much has M fallen in exchange- 
value in all other things, that is, in [A] and [B] together? 
Now the fall of M in [A] and [B] together is the same as its 
fall in [A] and [B] under the supposition that A and B both 
rose together to some extent. Therefore this problem is the 
same as to ask : To what rise of A and B together is the rise of 
A alone equal in its influence upon the exchange-value of M in all 
(both these) other classes of things f What we want is to reduce 
irregular variations in the particular exchange-values of the 
money-unit to a uniform variation of all of them, because when 
we have the latter, we know the variation in the exchange- value 
of the money-unit in all other things (by Proposition XVII.). 

§ 2. To this problem an answer might be made which has 
the nature of an objection. It might be argued that, though M 

2 Cf. also Proposition XXXIII. 



TIIK I'KOBLKM AM) SlcaJKSTED SOU'TIONS 215 

lias fallen from the power of purchasing A to the power of pur- 
chasing only |A, yet as this |A has risen by one half in ex- 
change-value in all other things, therefore M still commands as 
much exchange- value in all other things as before, and still ])os- 
sesses as much ; and with M also B. But this argument as- 
sumes that A has risen by one half in exchange-value in all 
other things themselves remaining constant in exchange-value 
in all other things. This implied assunij)ti()n is false. With 
the same evidence with which we })erceive that 2V has risen by 
one half in exchange- value in all other things, we perceive that 
the other things have fallen somewhat in exchange-value in all 
other things. We know that M has fallen as well as we know 
that A has risen. And the two statements are perfectly con- 
sistent. For ^vhen M purchases two thirds of A, which has 
risen by one half in exchange- value in all other things, them- 
selves fallen in all other things, M does not command so much 
exchange-value in all other things as it did before. 

Then another solution might at once be offered, which, if cor- 
rect, would settle our problem without further trouble. It 
might be argued that if at the second period j\I had become 
equivalent to i(|A -|- §B), it would have fallen by one third in 
exchange-value in all other things ; therefore, since it has fallen 
to equivalence with |(|A -f- B), it has fallen by one sixth, a 
fall of one third in one thing being equal to a fall of one sixth 
in two things. But this argument makes a mistake similar to 
that in the preceding. That argument neglected to take account 
of the fall of M and of B in exchange-value in all other things. 
This argument neo-lects to take account of the fall of B in ex- 
change-value in all other things. M purchases only one whole 
of an article which has fallen with it. It purchases two thirds 
of an article which has risen by one half in exchange-value in 
articles which have fallen. Therefore M would seem to have 
fallen by more than one sixth. But from these data alone no 
definite conclusion can be drawn concerning the extent of the 
fiill of M (and of B) in exchange-value in all other things. 

§ .3. If we could know how much A has risen in exchange- 
value in all things, this would give us sufficient information 



21(3 THE QUESTION OF THE MEANS AND AVERAGES 

from which to calculate the exchange-values of M and of B. 
But we cannot know that until we know how much the ex- 
change-values of M and B have fallen. Thus the beginning 
seems to be lacking. 

The trouble is, we are as yet in possession of only one equa- 
tion with two unknown quantities. We know the equality of 
the exchange-value of M at the second period to the exchange- 
value of f A and of B. But we do not know the exchange- 
value of B any more than we know that of M itself; and al- 
though we know the exchange-value of A in all other things, 
that is, other to it, this is in another standard, and we do not 
know the exchange-value of A in the standard of other things 
for M, namely [A] itself and [B]. We cannot know this 
until ^ve know the variation of B, or of M itself, in this stand- 
ard. A further piece of information is wanting. In order to 
solve the problem we must supply ourselves somehow with this 
other information. Further consideration of the question, with 
variations in the manner of putting it, ma}' perhaps yield us 
further insight into the nature of our subject, and so disclose 
that we are in possession of the information desired. 

§ 4. Xow we do possess a piece of information, already 
noticed, which is of some service here. This is the knowledge 
that the variation of M (and of B) in exchange-value in all 
other things, when A rises to 1^, is less than the inverse varia- 
tion from 1 to f , that is, it is to some quantity between 1 and 
-|.^ This provides us with a hint. Between two numbers we 
are acquainted with several mathematical means, and the idea 
suggests itself that the variation in question may perhaps be to 
one of the mathematical means between 1 and f . This sugges- 
tion becomes more plausible when Ave recall that the formulae we 
have discovered for exchange-value variations are expressions of 
the three " classic " means or averages. Thus if it be true that 
the variation of Mq., compared with Mq^ is to one of these means 

^ According to Proposition XVIII. Evidently to produce the same influence 
upon the exchange- value of M a common variation of A and B together cannot be 
so large as the variation of A alone ; for if we had the variation of A alone in the 
first place, influencing M to a certain extent, an added vai'iation of B in the same 
direction would increase the influence upon M. Compare with tliis tlie reasoning 
leading up to Proposition XXX. 



THK rnoBLEM AND SICUJESTKI) SOLTTIONS 217 

between 1 and |, it would be either to 9(0 + 1) = -(= .<S3o3), 

1 4 

the arithmetic, or to ^/q \ = f(= •'^*')? ^l^e harmonic, or to 

2(2 + ' 



1/ - • 1 = .8165, the geometric ; whence the percentage of the 

fall may be calculated, in hundredths, by subtracting the result 
from unity. To be sure, we have liere only a hint, but it is 
sufficient to incline us to try these suggested answers, and so at 
once to narrow our enquiries ; for, although there are other more 
com])lex mathematical means, we need not notice them unless 
the true answer be not found among the simpler ones.^ 

And tliis hint extends to the average of prices also. The re- 
lationship between these and the averages of the exchange- 
values of M we have already seen. Thus in our suppositional 
case the recij)rocal of the arithmetic mean between 1 and |, 
namely, .8333, is 1.20, which is the harmonic mean between 
1.00 and 1.50 ; the reciprocal of the harmonic mean between 
those exchange-values, namely .80, is 1.25, which is the arith- 
metic mean between these prices ; and the reciprocal of the geo- 
metric mean between those exchange- values, namely .8165, is 
1.2247, which is likewise the geometric mean between these 
prices. Therefore if we find to which average between 1 and f 
the exchange-value of M in all other things has sunk when A 
alone rises to 1^, we shall know to which average between 1.00 
and 1.5(.) the rise of both A and B together is, in its iufluence 
upon jNI, equal to the rise of A alone to 1.50. 

Thus our simple problem becomes a question of mathematical 

means. And as it has been stated in the simplest way ])0ssible, 

it has been made to em])loy single weighting. The question has 

been put as a question of means between variations of exchange- 

■'The reader cannot be too often warned that we are not averging |A and IB, 
but S.l and IB — we are not averaging, for instance, ?, pound of sugar and 1 pound 
of copper, which would be meaningless, but we are averaging the exchange- value 
of M in [A], say sugar, and the exchange-value of M in [B], say copper, in each 
case measured at the second period by comparison with what it was at the first. 
These exciiange-values, or these variations of exchange-values, are similar and 
co-ordinate things between which it is perfectly proper to draw averages. 



218 THE QUESTIOX OF THE MEANS AND AVERAGES 

values, or of prices. lu this form the problem may be extended 
to complex cases, embracmg more than two classes, and classes 
unevenly important. It then becomes a question of mathemat- 
ical averages, still with single weighting, between price varia- 
tions — for in these cases the variations of the exchange-values 
of money have generally been relegated to the rear behind the 
variations of prices. 

Now we may find that, while the question of the mean may 
be definitely answerable in the simple case posited, yet the ques- 
tion of the averages may not be in the complex cases, except on 
special occasions. It is plain that the above indication of the 
answer to our general problem is not necessarily complete. The 
universal solution may perhaps be yielded, not by a mathemat- 
ical average of the price variations, but by a comparison of 
mathematical averages of prices at the two periods compared, 
possibly requiring the use of double weighting. We at least 
see that we have these more complex solutions in reserve, in 
case the more desirable, because simpler, solutions, which first 
suggest themselves, shall fail. 

II. 

§ 1. The suggestion that the general exchange- value of 
money has changed to some average of the variations of its 
many exchange-values, or to the reciprocal of some average of 
the price variations, is so plain that it has mostly been acted 
upon without question. Indeed it was followed for more than 
a century before the problem itself was even definitely raised. 
And it was followed very specifically. Among the three aver- 
ages was selected the arithmetic, this being the simplest, the 
easiest to manage, the most familiar, and therefore the first to 
recommend itself. The adoption of this average has been due 
to no consideration of its special propriety for the subject it is 
applied to, but merely to a general preference for it in every 
case. This is shown by the way it has been applied. Statis- 
ticians have mostly conducted their investigations about varia- 
tions in the exchange-value of money simply by noting, variations 
in the prices of commodities ; in doing which they have oper- 



HISTORY OF THE (^rESTION "219 

ated upon the variations of tho exchange- values of commoditie.s 
in money, and not directly upon the variations of the exchange- 
values of money in commodities, but inversely upon the recip- 
rocals of these. It happened then that, dealing merely with 
prices, the early workers in this field applied the arithmetic 
average to the variations of ])rices ; and most of the later ones 
have followed suit. Thus they have unwittingly employed the 
most difficult and least familiar average, the harmonic, for the 
direct averaging of the variations of the particular exchange- 
values of money. Had they happened to conduct their en- 
({uiries directly upon these, as measured by the quantities of the 
things purchasable with the money-unit, it is likely they would 
still have employed the arithmetic average, Avhich then would 
be the harmonic average of the price variations, — and with 
equally plausible reason, if a reason be desired, since this aver- 
age is the one which above presented itself to us as the most 
specious at the first glance, when we looked at the subject from 
this side. In fact this course has actually been })ursued. In 
India it is customary to report, not the prices of corn, l)ut the 
numbers of seers purchasable per rupee ; ^ and in averaging 
these the arithmetic average has been used,^ And not long ago 
at least one statistical historian, M. I'abbe Hannauer, presented 
his facts in this form and applied to them the arithmetic aver- 
age, thus really using the harmonic average of the price vari- 
ations.^ 

While many writers were so engaged in arithmetically aver- 
aging variations, generally of prices, with single weighting, an- 
other set of writers arithmetically averaged separately the prices 
at each of the' periods compared, but, likewise using single 
weighting, on the assumption that the same mass-quantities Avere 
produced or consumed at all the periods compared. In doing 
so they employed a different method ; for the others, whether 
they used even or uneven weighting, made no such assumption 

^ See Prices and ivages in India, compiled in the Statistical Branch of the 
Department of Finance and Commerce, Calcutta 1885, (under the direction of J. 
E. O'Conor). Cf. D. Barbour, Theory of bimetallism, London 1880, p. 121. 

^ By Palgrave, copying from Charles C. Prinsep, B. 77, pp. 378-383. 

* See B. 3.5. Also D'Avenel, in B. 117, seems to have pursued the same course. 



220 THE QUESTIOX OF THE MEANS AND AYEEAGES 

about the mass-quantities. Yet none of the writers in these two 
sets seemed to recognize that they were pursuing two different 
(though convertible) lines of inquiry (the ones hailing' from Carli, 
the others from Scrope, — for the followers of Dutot do not de- 
serve further notice). And we have seen that both these sets 
have admitted many inconsistencies in their methods, consisting 
of kinds of ^^eighting never intended (among them even double 
weightmg) doing so because they carelessly did not examine what 
they were doing, and were led astray by mere convenience. 

§ 2. The merit of raising the question between the three aver- 
ages belongs to Jevons, who took up the subject in 1 8 63. Jevons 
belonged to the first of the above two sets — and avowedly so, 
for we have seen that he denied the propriety of averaging prices 
at any one period alone. He posed the question in somewhat 
the same way as has above been done, but very barely. * At 
first he does not appear to have been acquainted with the work 
of his predecessors, and, coming to the subject without prejudice, 
he stumbled upon the problem at the outset, and at once decided 
it in favor of the geometric mean and average (with even weight- 
mg), from which he M'as not repelled by its difficulty, because, 
being a mathematician, he knew the aid to be derived from the 
use of logarithms. He was very brief in explaining why he 
chose this answer, and the reason which determined him is only 
very slightly indicated. It seems to have been that variations 
of prices are variations of ratios, and the proper method of aver- 
aging ratios is always the geometric. ^ Within a year his posi- 
tion was assailed by Professor Laspeyres, who advocated the 
arithmetic average (still with even weighting), and now for the 
first time advanced an argument in its behalf. This argument, 
which we shall examine in detail later, was that the arithmetic 
average of the price variations marks the variation in the sum 
of money needed to purchase the same goods at the two periods 
compared. Jevons was not converted, and yet he sJiowed no 

* He asked merely : If one thing rises in price from 100 to 150 and another 
from 100 to 120, what is the average rise of prices? B. 22, p. 23. 

^ B. 22, pp. 23-24. Tliis reason was later more plainly stated in his Principles 
of science, 1874 (2d ed., p. 361), in a passage which will later be quoted (in 
Chapter YIII.) ; and again in a note added to B. 23, p. 128. 



irisTORV OF THE (^rj-:sTi(>N 221 

sijrn of (l('tcctint>' tlic error in Laspeyres's ar*runu'iit. Hv reaf- 
firmed his j)ositioii tlie next year, and now introduced notice of 
the liarnionic mean of ])riee variations, attain for the first time, 
whicii mean (with even weii»'htin^-), lie ])ointed out, marks the 
variation in tlic quantity of goods the same sum of money evenly 
distributed will purchase at the two periods. This average he 
set over against the arithmetic average, and not being able to 
see why either should be jireferred to the other, he found satis- 
faction in his geometric average as lying between those equally 
good extreme averages, and therefore combining the excellences 
of both. " He added : " It is probable that each of these is right 
for its own purposes when these are more clearly understood in 
theory." But instead of taking pains to analyze the subject 
further in order to reach the needed clearer understanding of it, 
he contented himself with the above reasons, to which he added 
the following : " because it [the geometric mean] presents facil- 
ities for the calculation and correction of results by the continued 
use of logarithms," — a worthless reason in science, and a foolish 
one to oppose to the much more facile arithmetic average. '^ 
Consequently he never felt sure of his position, and often spoke 
doubtfully about it, ^ and took refuge in the thought that at all 
events he would be erring on the safe side ; for he ^v•as writing 
to prove that there had been a general rise of prices, and there- 
fore he preferred to underestimate rather than to run any risk of 
overestimating it. ^ 

Following Jevons, the geometric average has been adopted 
by the eminent mathematical economist, Professor Walras, who, 
however, has found no better reason to oifer for it than Jevons's 
second reason (about its being midAvay between the other tw^o 
averages), even asserting this to be the only reason for employ- 

«B. 23, pp. 120-121. 

■^ This is given as the second in a summing up containing tliree arguments, 
ibid., pp. 121-122. The third is the first one above noticed, the original one. The 
first is the later one just explained. 

* " It may be a matter of opinion which result is the truer " (the geometric or 
the arithmetic), B. 24, p. 154. And he maintained similar reserve in regard to 
these averages, still looking upon the question as unsettled, in his Money and the 
mechanism of exchange, 1873, p. 332. 

^Cf. B. 23,' p. 122, B. 24, p. 154. This has sometimes l)een taken for his prin- 
cipal reason. It was, of course, an aftertliought. 



222 THE QUESTIOX OF THE MEANS AND AVERAGES 

ing it.^" Consequently he, too, has been weak in the faith ; and 
he has inclined more and more to prefer the arithmetic average 
in one of its forms, without recognizing it." But his choice 
has really lain only between the geometric average with even 
weighting and the arithmetic average in a form embracing un- 
even ^veightmg, so that it is only natural he should have pre- 
ferred the latter. With this restricted view he has not given 
the geometric average a fair trial, failing to perceive that it also 
can be used with allowance for the relative sizes of the classes. ^^ 
The cause of the geometric average being thus feebly pleaded, 
it has met with little favor, ^^ in spite of the high authority of 
its originator, who has even been treated with scant courtesy.^* 
More recently Professor Westergaard has advanced for it the 
argument which we have alreadv examined and found wanting. 
And occasionally Professor Edgeworth has put in a good word 
for it.^^ 

§ 3. Modern workers, then, have continued to make use of 
the arithmetic average of prices, or of price variations. They 
have done so commonly without regard for Laspeyres's or for 
anv other argument, properly so called, in its defense, ^"^ but, like 
the older ^^Titers, merely because it is the easiest to execute and 

"B. 69, p. 15. 

"See Appendix C, IV. §1. 

12 The geometric average as used by Jevons, with even weighting, has similarly 
been rejected, without notice of the possibility of its being used with uneven 
weighting, by Wicksell, B. 139, p. 8. 

13 The only other investigator who has actually used it is Forbes. See B. 78. 
1* Jevons's use of the geometric average was treated with levity by Paasche, B. 

33, p. 51, and as a curiosity by Lehr, B. (38, p. 41 n. Marshall, without argument, 
considers it " a mathematical error, the one flaw in his unrivalled contributions 
to the theory of money and prices," B. 93, p. 372 n.; Pierson pronounces it "clearly 
a mistake," p. 130 in the article referred to in note to B. 122 ; and Padan attacks 
it savagely, but with little comprehension, B. 141, pp. 171-180.— Walras's use of 
the geometric average seems to have been passed by unnoticed. 

15 The latter recommends it, with even weighting, in case we are seeking the 
" Determination of an Index irrespective of the quantities of commodities ; upon 
tlie hj'pothesis that there is a numerous group of articles whose prices vary after 
the manner of a perfect mai-ket, with changes affecting the supply of money," B. 
59, pp. 280-289. This seems to refer to cases when all prices vary alike, in which 
cases the weighting is indifferent. But in these cases the kind of average itself is 
also indifferent. In no other case do we want to seek any determination "irre- 
spective of the quantities of commodities." 

1*5 Laspeyres's argument is reviewed, favorably, by Lindsay, B. 114, p. 12, and 
by Mayo-Smith, B. 137, p. 492. 



IITSTOKY OF THE QUESTION 228 

because they have seen no proof of the superiority of any other. 
Moreover some experiments, by no means conclusive, show what 
has been deemed small divergence in the results yielded by the 
different methods. Hence it has seemed like a dictate of wis- 
dom to adopt the easiest, so long- as we do not know it to be 
wrontr.'" But it is not an exhibition of the proper scientific 
spirit to be content to remain in this ignorance. ^^ Beside Pro- 
fessor Walras, a few writers, after avowedly rejecting all three 
averages, have lighted upon the arithmetic average again, in one 
of its forms, without recognizing it.''' 

§ 4. Conseipicntly also the harmonic average of prices, which 
is the most difficult of all to manage, and the one to which 
people are the least accustomed, has met with the least favor, and, 
in fact, has generally been altogether ignored, even on the rare 
occasions it has virtually been applied. Yet, as we have seen, 
it is this average which is first likely to apjieal to us when we 
approach the subject from the point of view of the particular 
exchange-values of money in other things, measured by the 
quantities of the other things the money-unit will purchase ; 
where it corresponds to the arithmetic average. This reason, 
which still manifests a predilection for the arithmetic average, 
is the one which led Jevons to suggest the harmonic average of 
prices. And it has since led one ])rominent economist and 
statistician. Professor Messedaglia, to recommend its use in in- 
vestigations about variations in the exchange-value of money. 
This writer thought he showed the suitability of the harmonic 

1' Thus its retention is advised by J^dgeworth oil the principle that "beggars 
cannot be choosers," B. 05, p. ,S8(5. 

•* It should be noticed that in the usual method of comparing every subsequent 
period with a single original base, the divergences, which are no greater in the 
last than in the first comparison, are not of much consequence — and this whole 
method l)eing wrong, it hardly matters which average is used. But in the proper 
method of comparing each period with the immediately preceding and of forming 
a series from the results so obtained, a very slight error in every comparison, 
which might perhaps have a tendency to work in the same direction, caused by 
the use of a wrong average, could rapidly accumulate to an absurd extent. Hence 
the selection of the right average is of the utmost consequence. Having the right 
average, or the right method we should probal)ly make small mistakes in every 
calculation, but by the law of probability these would fall about equally on 
each side and would compensate one anotlier in tlie long run. 

' a See Appendix C, Y. ? 1 n. 2. 



224 THE QUESTION OF THE MEANS AND AVERAGES 

average of price variations for this purpose by pointing out — 
here like Jevons — that it indicates the common variation in the 
quantities of goods purchasable at the diiFerent periods with the 
same sum of money ; for it thus indicates the variation in the 
" capacity of acquisition," or purchasing power, of this sum of 
money. On the other hand, here like Laspeyres, he pointed 
out another purpose served by the arithmetic average of price 
variations, namely to indicate the common variation in the sums 
of money needed to purchase the same goods at the two periods ; 
only, unlike Laspeyres, he did not think this shows the suit- 
ability of this average for the previous purpose, for which it 
has ordinarily been used, as it indicates rather the variation in 
the capacity of goods to acquire money. The geometric aver- 
age of price-variations he thought to be unsuitable for either of 
these purposes, and not finding any other purpose for which it 
is suitable, he considered it, in our subject, good for nothing.-*^ 
Thus he has left the subject divided, with two distinct solutions, 
where Jevons tried to mediate and to give one solution. In do- 
ing so he has made no use of his knoAvledge that, in his own 
words, "the geometric mean . . . corresponds to a dynamic con- 
cept, of movement." ^^ As exchange-value is a power, or dy- 
namis, it is strange that his attention was not called to the pe- 
culiar appearance of fitness in the geometric average to serve 
our very purpose. 

It may be added that a few writers have tentatively put for- 
ward the so-called " median mean " and the " mean of greatest 
thickness." ^^ These are, properly speaking, not " means " at 
all, but only "averages," having no existence between two 
quantities. In an hypothesis concerning the mean between 
variations of only two prices, as they have no place, they do not 
call for attention. 

§ 5. While Jevons and Laspeyres were debating the merits of 

2 B. 52, pp. 38-40. 

2i/Wd,p. 30. 

2 2 The median mean is especially favored by EdgeWorth. He recommends it 
when we are seeking the "Determination of an Index based upon quantities of 
commodities ; upon the hypothesis that a common cause has produced a general 
variation of prices," B. 59, pp. 289-293. See also B. 61, pp. 360-363, and B. 65, 
p. 74. 



l!IS'r()|;\- Ol' '11 IK (IKKSTION -I'l') 

the <i('()iii('ti"ic, aritliinctic mikI liiinn(>iii(^ means ai)|)H('(l to juMce 
variations with even \vc'i<>litin<>', the economist Rosclier submit- 
ted the (juestion fur decision to the mathematician Drobiscli. 
Drobisch decided it by summarily rejecting all three of the means 
— although what he really did was to reject them with even 
weighting- for conditions demanding uneven weighting. He also 
did more. He reject(>d the averaging of ])rice variations with 
single weighting, and rejected the use of the same mass-quanti- 
ties at l)oth periods in avei'aging the ])rices at each period. In 
their place he introduced the method of double weighting — with 
the arithmetic average, and with what we have seen to be a mis- 
taken method of selecting the mass-units. Except for a vigor- 
ous reply by Laspeyn^s, this new method met with little notice, 
and was almost unknown outside of Germany. In Germany, 
however, an improvement upon it was later made by Professor 
Lehr. And in England another economist. Professor Nichol- 
son, apparently without knowledge either of Drobisch or of Lehr, 
invented a method very similar — in fact, in some aspects quite 
similar — to Drobisch's. These three, Drobisch, Lehr and 
Nicholson, form another distinct line of theorists, — differentiated 
fi'om the rest, however, rather on the subject of weighting than 
on the subject of the averages. 

The ])osition of these, the most difficult, and intended to meet 
the most difficult cases, we may leave to the last in our attempt 
to solve anew the problem above posed — a problem raised more 
than a third of a century ago and not yet settled. We may 
]>roceed circumspectly, beginning with the simplest and most 
easily suggested answers to the simplest form of the problem. The 
simplest and first suggested ans"\ver is that of a mean or average 
of the price variations, with single weighting. This we must 
examine first. If it foils us, — or when it fails us, for it may 
perhaps suffice for the very simplest form of the problem (with 
two classes and even Aveighting), — we must then pass on to seek 
other solutions on other lines — either with supposedly perma- 
nent mass-quantities or Avith double weighting. 

15 



CHAPTER VII. 

BRIEF COMPARISON OF THE MEANS. 
I. 

§ 1. Before examining- the arguments proper for the different 
mathematical means of the variations, it will be well to compare 
these means with one another. The comjjarison will not only 
make us better acquainted with them, but may perhaps even 
give some indication of the superiority of one of them by dis- 
closing shortcomings in the other two, and so aid our under- 
standing of the arguments. For this purpose it will be advisable 
to put our simple problem in another form. We may make use 
of another piece of information already possessed. This is that 
if one article rises or falls, its variation may be compensated by 
an opposite fall or rise of another article. As yet we do not 
know how great must be the opposite variation. Here also is a 
Proposition lacking, which we wish to find. The seekmg after 
it is the same operation as the seeking after the missing Prop- 
osition referred to in the last Chapter, as will be seen in a 
moment. Keeping to our simplest suppositional case; we may 
ask this question : — 

Among three classes of things, two being equally important, 
whose units, M, A and B, are at first equivalent, if one of them, 
A, rises m exchange-value in [M] by one-half, so that its price, 
from being 1.00, becomes 1.50, and so that M, from purchasing 
A, comes to purchase only f A, how much of B must M pur- 
chase in order that its exchange-value in [A] and [B] together 
shall remain as it was before, and to what price, from 1.00, will 
the price of B then fall ? 

Here the new exchange- value of M in [A] and the new ex- 
change-value of M in [B] will have the old exchange-values of 

226 



ANOTHER FOUM OF THK l'HOIU,EM 22 < 

M in [A] and in [B], which were nnits, between them, and 
therefore possibly as a mean of some sort ; and consequently the 
new price of A and the new i)rice of B will have their old ])rie('s, 
1.00, between them as a mean of a correspond inir sort. Only 
instead of seeking the mean between two given extremes, we arc 
seeking the absent extreme when one extreme and the mean are 
given. Hence this problem is the same as the ])roblem set in 
the last Chapter. 

In this problem we shall for the present assume that the classes 
are equally large or important, not at either of the periods alone, 
but at each of the periods, or somehow over both the ])eriods. 
But we shall attempt to leave the question of weighting out of 
sight so far as possible. We have already treated the question 
of Aveighting by itself, as far as we could ; and now we shall 
try U) treat the question of the averages by itself, as far as we 
can. We shall, however, find, as we proceed, that we can not 
settle this question any more than that question separately. We 
shall then have to combine the two into one question about the 
variation of the general exchange-value of money under given 
conditions of variations both of prices and of mass-quantities. 
This course is adopted because it re})eats the usual way the sub- 
ject has been approached, although only three writers have as yet 
carried it to the end. In pursuing it all opinions hitherto ad- 
vanced may be reviewed. 

§ 2. The problem in its new form, may, for the exchange- 
values of M, be formulated in our three formulae, in the order 
with which we are familiar, as follows : 

M ^[ — 

^■^01 -1/1 1 \ - ^'^"' ~ 1 / 1 i_ y 



3f^, = l/ll =Mo^ = V r^h' 



/2" 
3 



For these formulae to be carried out, that is, for Mf,, to be equal 



228 BEIEF COMPAEISOX OF THE MEAXS 

to Ifoij we want the resultants of the figures in the second halves 
to equal those in the first halves, which are all units. We see 
that this will be the case in the first formula if 6' = 1|, in the 
second if b' = 2, in the third if 6' = 1 J. Thus if we later find 
any one of these figures to be the right answer, we shall know 
that its formula is the proper one, in similar cases, for averag- 
ing particular exchange-values of M in all other things at two 
periods for the purpose of comparing its exchange- value in all 
other things at those periods, as it alone gives the right answer 
in this simple case. 

The corresponding formula for the prices are, in the same 
order : 

P_^^^ =P= ^ 



2Vl.00'^1.00y 2V1.50"^;3' 

p^= 1(1.00-1- 1.00) = p. = 1(1.50 + ^'; 



P, = 1/1.00x1.00 =P2= t/1.50/3'. 

And similarly we want the results in the second halves to equal 
unity, which will be the case in the first formula if fi' = .75, in 
the second if ^9' = .50, in the third if /?' = .66f ; which are the 
reciprocals of the preceding answers for b' , in the same order. 
That one of these formulae for averaging prices, if any, will be 
the proper one, in which the answer for /?' is the reciprocal of 
the proper answer for b'. 

An advantage in re-stating the question in this form is that 
the diversity of the proifered extremes is greater than the di- 
versity of the proffered means. Also it brings in the idea of 
balancing, in which the idea of equality of influence is plainer 
than in the case of conjoint action. The review and comparison 
of the suggested answers will now disclose interesting relations. 
And two of the answers will manifest unmistakable signs of 
falsity. 

II. 

§ 1. Only the purchasable quantities of the articles being 
considered (which are directly according to the exchange-values 



THE COMPARISON 229 

<)f M in tlu'in), it is easy to answer that if M will purchase only 
I as uiii(!li of [A] as before, it should purchase J more of [B] 
in order to make up for the ^ lost on [A] . Here the new cpian- 
tities purchasable with M (and the new exchange-values of M 
in [A] and [BJ ) surround the old (piantities (and exchange- 
values), which wei'e units, so as to hold them in the arithmetic 
mean, the progression being |, 1, 1^^. The new prices, then, 
are 1.50 for A and .75 for B, which hold the old prices, 1.00, 
as their harmonic mean, in the progression 1.50, 1.00, .75. The 
proper formulse, if this answer be correct, are 

^^ = 2(3 + 4) ="^^^ ^^ = 1/^' ^Y 

2 VI. 50 "^ .75J 

the result in each of these being unity. 

Only the prices being considered, it is easy to answer that if 
A has risen to 1.50, B should foil to .50, so that the increase of 
money needed to purchase A shall be made up for by the de- 
crease of money needed to purchase B. Here the new prices 
have the old ])rices as their arithmetic mean, the progression 
being 1.50, 1.00, .50. On examining the new quantities pur- 
chasable with M, Ave find that they are f A and 2 B, which 
have the old quantities, which were units, between them in the 
harmonic mean, according to the progression |, 1, 2. The 
proper formula for this answer are 



■2\2i 



13 + 

Both the quantities and the prices being considered, it is pos- 
sible to treat them uniformly by answering that if the price of 
A rises to ^ times its former position, the price of B should fall 
to f of its former position, and that if the quantity of [A] pur- 
chasable with M falls to | of its former amount, the quantity of 
[B] purchasable with INI should rise | times its former amount, 
— whereby are jn'oduced the same variations in the exchange- 
values of A in [1\[] and of B in [M] as in the exchange-values 



230 



BRIEF COMPAEISON OF THE MEANS 



of M in [B] and of M in [A] . Here both the new prices and 
the new quantities have their old prices and their old quantities, 
which were units, between them as geometric means, in the pro- 
gressions |, 1, |, and |, 1, |. The proper formulae are 



Mf,2 =1 f • U and P^ = 1/1.50 x .66f. 

§ 2, Thus, so to speak, the harmonic and arithmetic aver- 
ages are locked in an inseparable embrace, but the geometric 
stands by itself, self-sufficient. And both its answers are mid- 
^vay between those of the others. The peculiarities of these 
three methods of answering may best be shown by drawmg up 
a table of several supposed variations in price of A and of the 
counterbalancing variations in price of B according to the three 
methods, and paralleling them with the variations in the quan- 
tities purchasable with M. 







Counterbalancing 


2=" 





Counterbalancing new 


l< 





new prices of B accord- 




quantities of B 


accord- 


SK 


ing to progressions : 


ri 




ing to progressions : 




.S'^ 






'i^a 






«=> 














|S 


art 


5.2 


e^ 


1 6 


as 


%< 


1 ^ 

S'3 


1 


5-2 


5*=^ 





< S 


■^l 


«g 









2 


<^ 


1.25 


1.00 


.75 


.80 


.831 


.8 




1.331 


1.25 


1.2 


1.331 


1.00 


.661 


.75 


.80 


.75 




1.5 


1.331 


1.25 


1.50 


1.00 


.50 i .661 


.75 


.661 




2 i 1.5 


1.331 


1.75 


1.00 


.25 i .571 


.70 


.57f 




4 1.75 


1.42f 


1.90 


1.00 


.10 


r;9l 2 


.67f 


.52H 




10 , 1.9 


\Alj\ 


2.00 


1.00 





.50 


•66f 


.5 




c» 2 


1.5 


2.50 


1.00 




.40 


.621^ . 


.4 




2.5 


1.6 


3.00 


1.00 




.33i 


.60 


.331 




3 


1.66| 


4.00 


1.00 




.25 


.571 


.25 




4 


1.75 


5.00 


1.00 




.20 


.55f 


.2 




5 


1.8 


10.00 


1.00 




.10 


.52H 


.1 






10 


1.9 


100.00 


1.00 


.01 


•50fV9 


.01 






100 


1.99 


00 


1.00 





.50 







1 00 


; 2 



On looking over this table we see that as the price of A is 
supposed to rise and the quantity of [A] purchasable with M 
to fall, the counterbalancing fall of the price of B according to 
the arithmetic progression is so rapid, and also the correspond- 
ing rise of the quantities of [B] purchasable with ]M at these 
prices, which quantities are according to the harmonic progres- 
sion, that when the price of A reaches 2.00 and thereafter there 
is no counterbalancing price of B, which would have to be zero 



Tin-; COMTAIMSON 2ol 

or loss, and no counterhalancinti' (|iiantity of B purcluisablc witli 
M, as it would have to be iuliiiite or becomi' a ne<^ativt' (|iiaii- 
tity, which hero is meaningless. And correspondingly, it" we 
consider the fall in the price of B and ask for the counterbalanc- 
ing rise in the price of A according to the harmonic progression, 
with the quantities in arithmetic ])r()gession, we find this rise so 
rapid as to reach infinity when the price of V> falls to .o(), and 
thereafter it would be a negative (|uantity,Avhic]i again is meaning- 
less here. In practice, however, these difficulties would not al- 
ways be insuperable, as we should then be dealing with many ar- 
ticles, and the compensation might be obtained by distributing the 
countorl)alancing change over two or more of them. Thus in the 
case of the arithmetic solution of prices, a fall of ])rices of 100 per 
cent, or greater could bo obtained by dividing it into several lots. 
Yet a rise of 100 per cent, on an average over half the number 
of articles could not be counterbalanced by any fall of the others. 
In the case of the harmonic solution of jirices, the rise to infin- 
ity would not be required if two or more articles rose in price, 
bi'cause a whole or greater (|uantity purchasable with ^I may 
bo lost by being distributed over two or more articles without 
any considerable rise of their prices. For instance, the fall of B 
to .50 can l)o compensated by the rise <»f two articles to 2. Of). 
Yet again the fall of half the numV)er on an average to loss than 
.oO could not be counterbalanced by any rise of the others. 
Those groat changes of prices on the part of a majority of com- 
modities are not to be expected unless money bo varying in its 
general exchange-value, and so the compensation may not be 
})ractically needed. Yet theory would seem to require its possi- 
bility always. . The geometrical method satisfies this theoretical 
requireraont. For every rise or fall of one article it ])r()vidos a 
Counterba lancing; fall or rise of another. 

^ '^. This tendeucy of the arithmetic and harmonic solutions 
to run into the ground or to fly into the air by their excessive 
demands is clear indication of their falsity. Error is often made 
})atent by examining extreme cases. It is so here. Take the 
case when A rises in price to l.!)0. You can then buy with 
one money -unit ten nineteenths as much of [A] as you could 



2."] 2 BRIEF COMPARISON OF THE MEAXS 

before, or a trifle over half. The arithmetic answer says you 
ought then to buy of [B] ten times as much as before. Thus, 
losing nine nineteenths or not quite half on [A] , you gain nine 
wholes on [B] . The excessiveness is evident. Your monev 
has appreciated. On the other hand consider the fall of the 
quantity of [B] to .52i|. With your money-unit you can now 
get nine tenths more of [B] than before, or not quite double. 
Yet the harmonic method of calculating the terms of prices would 
require the price of A to rise to 10.00, which will enable you 
with your money-unit to get only one tenth so much of [A] as 
formerly. Again the excessiveness is evident. Your money 
lias depreciated. The geometric method commits neither of these 
excesses. In the former case it does not require the price of B 
to fall so low as did the arithmetic — only to .ooi| instead of to 
.10; and it enables you then to get nineteen tenths instead of 
ten wholes, so that, losing nine nineteenths on [A] , you gain 
nine tenths on [B] instead of niue wholes. There is no appear- 
ance here of appreciation. In the latter case, this method does 
not require the price of A to rise so high as did the harmonic — 
only to 1.90 instead of to 10.00 ; and it enables you then to get 
with your money-unit ten nineteenths instead of only one tenth 
of [A] , so that, gaining niue tenths on [B] , or not quite as 
much again, yon lose nine nineteenths on [A] instead of nine 
tenths, or nearly the whole. Neither is there here any appear- 
ance of depreciation. There is thus an appearance that the 
geometric mean may have rendered your money stable in general 
exchange-value, since it has removed the appearance both of 
appreciation and of depreciation. 

As yet, however, we are but slightly advanced beyond Jevons 
in one of his moods. Jevons acknowledged himself at a loss 
between the opposite merits, as he admitted them, of the arith- 
metic and harmonic methods of averaging prices, and rested con- 
tent with a compromise on the intermediate geometric method. 
We see the ojiposite failings of those methods, and welcome the 
geometric method for avoiding each extreme, and so suggesting 
at least the possibility of its being the correct one — under the 
assumed conditions at least. 



CHArXER YIII. 

THE GENERAL AR(;UMEXT FOR THE GEOMETRIC MEAN. 



§ 1 . The problem before ii.s has two aspects. The one is the 
|)()int of view of prices, the other the point of view of the ex- 
chaiitic- values of money in the g'oods priced. Of two equally 
important classes of goods the rise of the price of the one is to 
be compensated by a fall of the price of the other ; and the rise 
of the exchange- value of money in the latter is to be compen- 
sated by a fall of the exchange- value of money in the former. 
Tlie second is really our main problem ; but its place may be 
tak(ni l)y the first, for the- sake of convenience. No argument, 
however, is good showing the superiority of one kind of com- 
pensation in the })roblem of prices, unless it can show also the 
superiority of the kind of compensation thereby involved in the 
l)roblem of the exchange-value of money. In order, then, to be 
able to apply the question of compensation and of averages to 
our dt)uble-faced problem, we must understand something of the 
nature of compensation iu general, and of the nature of averages 
in reference to the kinds of subjects for which they are generally 
suitable. 

In the question of compensatory variations, the following 
])roposition at once strikes us as evident : — A eompensatory varia- 
tion ill the one direction mud equal the variation in the other direc- 
tion which it is to conipenm-fe. 

We generally treat variations by percentage, and we have an 
habitually established manner of reckoning percentages. We 
treat an increase or rise on the one hand, and on the other a de- 
crease or fall, as variations from one common point or position, 
or whole, arbitrarily chosen as that which exists at the com- 

233 



234 GENER^vL ARGUMENT FOR THE GEOMETRIC MEAX 

mencement. The percentage of the rise is the ratio of the later 
excess beyond the original whole to that whole, multiplied by 
one hundred ; and the percentage of the fall is the ratio of the 
later deficiency below the original whole to that Avhole, multi- 
plied by one hundred. Hereupon we are inclined to maintain — 
and this is, in almost all cases, the fundamental idea in the argu- 
ment for the arithmetic average of variations — that a compen- 
satory fall should always have the same jpercentage as the rise it is 
to compensate, and reversely. But the above is not the only way 
of reckoning percentages. 

Of reckoning percentages there are three different ways — or 
better, two distinct ways, and a third constructed by uniting 
them ; wherefore there fire three different ways of obtaining 
sameness of percentage in compensatory variations. Conse- 
quently the argument for the arithmetic average has no validity 
unless it can be proved that the ordinary way of reckoning per- 
centages is the proper way of reckoning them in the special case 
in question. The ordinary way of reckoning percentages, or of 
measuring variations, is the way we shall always continue to 
pursue, because of its greater convenience. But unless it be 
the proper ^yay for indicating the equality of variations in the 
problem before us, we shall have to adapt it to our subject by 
adapting whatever turns out to be the _proper method of meas- 
uring the variations to this convenient method of measuring 
them. 

We must therefore examine the three ways in which we can 
conceive of equality in compensatory variations. 

§ 2. A being .supposed to rise in some attribute to a certain 
extent, equality to that rise in the fall of B in the same attribute 
may be represented in the followmg three ways : (1) B may 
fall so that the point to which it falls is as much below the point 
from which it falls as the point to which A has risen is above 
the point from which A has risen ; (2) B may fall so that the 
point from which it falls is as much above the point to which it 
falls as the point from which A has risen is below the point to 
which A has risen ; (3) B may fall so that either the point to 
which it falls is as much below the point from which it falls as 



HQUALITV IN < XM'OSrrK VA hMATfONS ' 24-) 

the point from which A has i-iscii is below the ))oiiit to wliicli A 
lias risen, oi" the point from wliich it falls is as lunch above the 
point to wliicli it falls as the point towhieh A has risen is above 
the point from which A has risen. In the first we nieasnre 
the variations, or their percentages, in the usual way, in what 
the thin<j;s vary from. In the other two we depart from the 
nsnal method. In the second we measure the percentages in 
what the things vary fo. In the third we measure the varia- 
tions either in what the one thing rises to and the other falls 
from, or in what the former rises from and the lattin- falls fo. 
In the first two we make each measurement in an o])])osite di- 
rection — in the first from the start, at the mean, to the ex- 
tremes ; in the second from the extremes at the starting points 
to the mean, — the former being centrifugal, so to speak, and 
the latter centripetal. In the third we make both measni-e- 
ments in the same direction, either both npwards or both down- 
wards, the starting point of the one being at the mean and that 
of the other at an extreme, or that of the hitter at the mean 
and that of the former at the other extreme. 

When the variations in Avhich e(|nality is obtained in these 
three ways arc all measured in the usual way, the first are \ari- 
utions from the mean to the simj)]e ' arithmetic extremes around 
it. Thus, for example, A rising from 1.00 to 1.50 by oO |)er 
cent, (reckoned in 1.00), B falls from 1.00 to .50 by 50 percent, 
(also reckoned in 1.00). They may therefore be called simple 
arithmetic, variations. The second arc variations from the mean 
to the simple hai-monic extremes around it. Thus, A rising 
from 1.00 to 1.50 so that 1.00 is o.'U per cent, below 1.50 
(reckoned in 1.50), B falls from 1.00 to .75 so that 1.00 is 'X\^ 
per cent, above .75 (reckoned in .75). They may therefore be 
called simple harmonic variation.^. The third are variations from 
the mean to the sim})le geometric extremes around it. Thus, A 
rising from 1.00 to 1.50 by 50 per cent, above 1.00 (reckoned 
in 1.00), B falls from 1.00 to .(>()§ so that 1.00 is 50 per cent, 
above .665 (reckoned in .<)(j|) ; or, A rising so that 1.00 is 3.')^ 
per cent, below 1.50 (reckoned in 1.50), B falls fnun 1.00 to 

* /. e., with even weighting, only two figures lieing usL'd. 



2o(J GEXEEAL ARCtU:MENT FOE THE GEOMETRIC MEAN 

.()6f SO that .66f is 33^ per cent, below 1.00 (reckoned in 1.00). 
They may therefore be called simple geometric variations. '~ 

Thus when we use in all the cases the ordinary method of 
reckoning the percentages, the equality exists only in the first 
one. Yet this equality is no more real than the equality which 
appears in the other ways of reckoning the percentages. 

§ 3. The equality manifested by the arithmetic variations 
seems to be of a peculiar nature. This is equality of diiference, 
or of distance traversed, 1.50 and .50 being equidistant, so to 
speak, from 1.00, wherefore A in rising from 1.00 to 1.50 and 
B in falling from 1.00 to .50 have traversed equal distances. 
This property may be thought, at first sight, to belong only to 
the arithmetic variations. It does so belong only to them if the 
variations must be conceived as startmg from 1.00, or from some 
common figure, that is, from the mean. But there is no reason, 
except mere convenience, why in all cases the variations must 
be so conceived. If they are conceived as starting from other 
figures, this property of traversing equal distances may belong 
to the other variations. 

Thus if A rises from 1.00 to 1.50 and B falls from 2.00 to 
1.50, A has risen by 50 per cent, and B has fallen by 25 per 
cent. Evidently the variation, merely as a variatiou, of J B 
from 1.00 to .75 is the same variation as that of B from 2.00 to 
1.50. Here we have harmonic variations. Yet the distances 
traversed are equal. 

And if A rises from 1.00 to 1.50 and B falls from 1.50 to 
1.00, or, B foiling from 1.00 to .66f, if A rises from .66| to 
1.00, A has risen by 50 per cent, and B has fallen by 33 J per 
cent., which are geometric variations ; for A has risen to be 50 
per cent, higher than it was, and B was 50 per cent, higher than 
it has come to be. But A and B have each traversed an equal 
distance — in fact, so to speak, the same road, only in opposite 
directions, so that m this case sameness is added to equality. 

These pro]>erties are universal. All variations from the same 
starting point to equal distances on opposite sides, ab( , ^ and 
below, are arithmetic variations (variations to arithmet. ^ ^^- 

- The universality of these rehitions is demonstrated in Appendix B, I. 



Kc^UAMT^' IX oi'i'osrn; \Ai;iA'i'i()Ns 2.'>7 

tremes) ; jukI all arithmetic variations may be reduced to such. 
All variations from equal distances on o])posite sides, above and 
below, to the same ending ])oint are harmonic variations (vari- 
ations to harmonic extremes, when measured in the usual way) ; 
and all harmonic variations may be reduced to such. All vari- 
ations over the same distances, between the same extremes, in 
opposite directions, upwards and downwards, are geometric 
variations (variations to geometric extremes, when measured in 
the usual way) ; and all simple geometric variations may be re- 
duced to such.' 

§ 4. Thus in all these variations we find two kinds of equal- 
ity : equality of ])r()portion, and equality of distance traversed, 
or of ditference. These may be represented algebraically as 
follows. I^et (t^ and a,, represent the figures at which A stands 
at the beginning and at the end of its variation, and 6, and b., 
the similar figures for the variation of B. Then in the arith- 

• • 1 ^' ~ "\ ^'\ ~^'> ^ -r 1 

metic variations we have " ^^^ ^ ^ , ', and \\a.= n,, a , — a, 

7 / • 1 1 • • • "■' ~ '^'l ^] — ^O 1 . . 

^ />, — 1), ; ni the liarmoinc variations, ' = — , ".and if 

(I., 0^ 

.«2 = /-'.„ ('., — <'i = f\ — f>., ; iiiid in the geometric variations, both 

a — rt, b — b., 

= ^^— ", mid if (I. = o.„ a., — (I, = f), — />, (whence also 

«., := b,), and -^— ' = ' - and if a, = b.. <i„ — a, = b, — b„ 

L \n ^,^ ''l - 1' - . 1 - 

(whence also a^ = />„).■* 

When one or both of these kinds of equality are present, as 
they may a])i)ear in three different ways, it may be well to give 
to these distinctive ap])ellations. Thus when we have the 
equality which ap])ears in arithmetic variations, we may name it 
arithmetic equnHtij of variations, or of percentages. When we 
have the equality Avhich apjiears in harmonic variations, we may 
name it harmonic equalitij of variations, or of percentages. And 
when we have the equality which appears in geometric varia- 



the universal demonstration see Appendix B, II. 
xJiese formulae make it clear that the above-used combinations exhaust all 
the possible ways of reckoning percentages and of getting equality between varia- 
tions. 



2o8 (iP:XERAI. ARGUMENT FOR THE GEOMETRIC MEAX 

tions, we may name it geometric equality of variations, or of 
percentages. 

Of the two general kinds of equality wliicli may run through 
all these varieties of variations, the equality of proportion must 
always be present if the variations are to be as we have been 
describing them (?'. e., if their percentages are to be equal in the 
three ways described). But the equality of diiference, or of dis- 
tance traversed, need be present only under certain conditions, 
different in each of the three varieties. Of course when we 
have two variations presented merely as variations, we m^y 
twist about the starting and ending points as we please, and so 
whenever we have an equality of proportion, we may also con- 
ceive of a corresponding equality of difference. If, however, the 
variations are ffiven us as the definite variations of an attribute 
in certain things, which have started from certain figures and 
have ended at certain figures, then an equality of proportion 
may be present, in either of the three forms, without the corre- 
sponding equality of difference, although the latter cannot be 
present without the former. The conditions for the presence of 
the equalities of difference are plain. In the case of arithmetic 
variations, the two things, or individuals, must be equally large 
(or important) at the first period, and at this period only ; for 
they both start from unity, or from any other common figure, 
that is, from the same level of equality in size or amount. In 
the case of harmonic variations, the two things, or individuals, 
must be equally large (or important) at the second period, and 
then only ; for they both end by being equal to unity, or to 
some other common figure, that is, they come to the same level 
of size. In the case of geometric variations, the two things, or 
individuals, must be alternately equal in size, the one at the first 
period with the other at the second, and again the former at the 
second with the latter at the first, so that they are equal in size 
over both the periods together. The connection between these con- 
ditions and the subject of weighting is obvious. 

When we have these equalities not merely of proportion but 
of distances actually traversed, we virtually have three different 
kinds of compensation — arithmetic, harmonic and geometric com- 



i-;(^r.\Li'i '^' in oppositk vaima'i ions 'I'M) 

jM'ii.sdfio}!. 'riicrc iiuiy \)v aiircenicnt between tlie.se at times, us 
M'e sliall see, and also disagreement. In the latter case it is the 
j)r()l)lem before us to decide which of these c(tni])ensations is the 
])ro|)cr one for our subject. 

S ~>. So far we have been dealing with (mly two things (or 
classes) that vary, and all these relations have been found to be 
universal when we are dealing with two things (or classes), with 
even weighting in the averaging of their variations. All of 
them are, or may l)e, universal, when unity is the arithmetic, 
hniMnonic or geometric iiieai). between two op})ositc variations. 
Furthermore, now, they are completely and unconditionally uni- 
versal in the cases of the arithmetic and harmonic averages, no 
matter ho^\' many variations on the one side be opposed to no 
matter how many on the other. In these cases if unity be the 
arithmetic or the harmonic average (in each case with the weight- 
ing of the ])eriod proper for it) between all the variations, there 
is ecpiality of proportion between all the variations on the one 
side together and all the variations on the other side together, 
and, with observance of the proper starting and finishing points, 
there is equality between the distance traversed by all the rising 
things and the distance traversed by all the falling things, the 
whole distance in each case being obtamed by summing up the 
distances traversed by all the things individually. 

Thus in the case of com])ound arithmetic variations, if in- 
stead of one B we have two B's, and both fall together from 1.00 
in the compensation for the rise of A from 1.00 to 1.50, the 
figure to which they fidl must be such that the arithmetic 
average between it twice repeated and 1.50 is unity. This 
figure is .75 ;■ for 1^(1.50 4- 2 x .75) = 1.00.'' Or if two things 
fall unequally, suppose the one has fallen to .80, then the other 
must fall to a figure between which and .80 and 1.50 the arith- 
metic average is unity. This is .70 ; for Kl-''^0 + -'^^ -|- .70) 

"' It should be noticed that the two figures, 1.50 and .75, are the simple harmonic 
extremes around 1.00, but the three figures, 1.50 and .75 twice repeated, are 
arithmetic extremes around 1.00. The figure .75 repeated a fractional number 
of times may even be a geometric extreme opposed to 1.50 ; for we may have 

' ^y^\.a(y7^^ = 1 .00, and it is not diflicult to find .r Mt is = ,^~f, = 1 .408(i | . 



240 GENERAL ARGUMEXT FOE THE GEOMETRIC MEAX 

= 1.00. And similarly in other cases.^ Here the sums of the 
percentage of the two falls, namely 25 + 25, and 20 + 30, are 
equal to the percentage of the one rise, namely 50 ; and the 
sums of the distances traversed by the two falling things, namely 
.25 + .25, and .20 -f .30, are equal to the distance traversed by 
the one rising thing, namely .50. These equalities exist — the latter 
if all the variations start from the same figure — whatever be the 
number of the arithmetically compensating variations on either 
side. The general formula with n equally rising and n' equally 

falling things is n ( — ^ ) = n' \ -^— y ) : and if a, = />,, 

n (a.-^ — ttj) ^ n' (6^ — 62). 

For compound harmonic variations, in a similar supposition, 

the formula is n ( — ~ ] = n' ( —-. ? | ; and if a„ = 6,, 

n [a^ — a J = n'(b^ — 62). For example, the rise of A alone from 
1.00 to 1.50 is harmonically compensated by the equal falls 

of two B's from 1.75 to 1.50 ; for ^ . ^ ^^ ^ ^^ , = 1.00. 

^ ^ \00 , 1.75 ^ 



1 /l.O 
3 V 1.5 



-f 2x 



3 V1.50 ' 1.50 

Or if one thing has already fallen from say 1.80 to 1.50, another 

must fall from 1.70 to 1.50 ; for 



1 /l.OO 1.80 



3 V 1.50 "*■ 1.50 "*" 



1.70\ 
1750 j 



= 1.00. And so in other cases. '^ Here, too, the sums of the 
percentages of the two falls, measured from the ending point, 
which are 16f + 16f, and 20 + 13^, are equal to the percen- 
tage of the one rise similarly measured, which is 33J ; and the 
sums of the distances traversed by the two falling things, 
namely .25 -f- .25, and .30 + .20, are equal to the distance trav- 
ersed by the one rising thing, namely .50. And these equalities 
exist — the latter if all the variations end at the same figure — 
whatever be the number of the compensating variations on 
either side. 

^ If the one has already fallen to .50, the other must remain at 1.00. If the one 
has already fallen beyond .50, the other must rise. 

■^ If the one has already fallen from 2.00 to 1.50, the other must remain at 1.50.^ 
If the one has fallen from above 2.00, the other must rise to 1.50 from below. 



K(iUALITV IN OPPOSITE VAKlATloXS 241 

i^ (). liut in the (M)mj)(>iin(l geometric variations tlie e(j[uality 
of |)roj)()rtion (!eases to exist, and the e((nality of distances trav- 
ersed exists only at tlie sa(;rifiee of some of the rehitions pre- 
viously included, so that we no longer have the same universal 
relations as before. This becomes evident the moment we deal 
with two equal variations in compensation for one variation. 
Thus the rise of A from 1.00 to 1.50 is geometrically compen- 
sated by the falls of two B's from 1 .00 to a figure between which 
twice repeated and l.oO the geometric average is unity. This 
is .8165 (=l/l; for ^/.r50 x .81652= 1.00. Here the 
percentage of the fall from 1.00 to .8165 reckoned in 1.00 is 
1 8.35, and twice this is more than the percentage of the rise of A 
reckoned in 1.50, which is 33^ ; and the percentage of the fall 
reckoned in .8165 is 22.47, and twice this is less than the per- 
centage of the rise reckoned in 1 .00, which is 50. As regards 
the distances tra\'ersed, the variations being kept the same 
merely as variations, the compensation may be by the two B's 
falling from 1.2247 to 1.00, or from 1.50 to 1.2247. Then in 
the former case they together traverse a shorter, and in the latter 
a greater, distance than A. Now if the one thing falls from 
1.50 to 1.2247 and the other from 1.2247 to 1.00, they both 
together traverse the same distance as A, covering in two stages 
downwards the same road covered by A in one leap upwards. 
But then the two B's are not equal things belonging to the same 
class (they are rather B and C, and unequal in size over both 
the periods together so that we should not be justified in geo- 
metrically averaging the three variations with even weighting). 
It is, however, possible for the two B's to be alike, and falling 
together in the same proportion as before, to traverse each half 
the distance traversed by A. This is when they both fall from 
1.3623 to 1.1123. But then the two B's are not each equally 
large or important with the one A over both the periods to- 
gether ; for the geometric mean between their starting and 
ending points is 1.23096, while the geometric mean between 
the starting and ending points of A is 1.2247. Thus the rela- 
tions that hold true of .simple geometric variations — variations 
of two equally important things (or classes), with even weight- 
16 



242 GENERAL ARGUMENT FOR THE GEOMETRIC MEAN 

ing, SO that unity is the geometric mean between them, do not 

hold true of compound geometric variations — variations between 

several opposed things, with various weighting, such that unity 

is a geom ric average between them. What has above been 

called geometric equality of variations or of percentages exists 

only between opposite variations of two things equally large or 

important over both the periods, and does not exist in more than 

two variations that geometrically compensate for one another 

(unless they do so in pairs). That this is so in general may be 

seen irom the formulae. With several things, n in number, 

falling together in compensation for the rise of A alone, the 

a„ — a, b"— b^ , . i . -, -n 

formulae are either^ = -^-7 , which is true only 11 a, 



or 



= 6, = 1, whereupon a^ — a^ = b" — b^, and a., = b" ; 

-^ = —-' — -- which is true onlv if a, = b. = 1, whereupon 

a^ — a_^ = 6j — b^", and a^ = 6^". Now while A traverses the 
distance a^ — a^, none of the opposing things traverse the dis- 
tance 6j" — b.^, or b^ — b^", but only the distance b^ — 6,, which 

is not - the distance a„ — «,. Still, trial shows that in ordinarv 

n i ^. ' 

cases, that is, with not more than two or three things opposing 
any one thing, the relations between compound geometric vari- 
ations are very nearly the same as those above described. 

What is here discovered, in differentiating the geometric 
average from the geometric mean, and segregating it from the 
arithmetic and harmonic averages, which behave universally m 
the same manner as do the arithmetic and harmonic means, will 
later call for our repeated attention. It shows that we cannot 
treat geometric averages as we do geometric means, and that the 
use of only two figures, with even weighting, will lead us into error 
if we treat them as examples good for all cases. After examin- 
ing what is true of the geometric mean, with two things evenly 
weighted, we must always turn to more complex examples ; and 
in them we shall find that convenient relations discovered in the 
former case no longer hold — except, in ordinary cases, only ap- 
proximately. We thus have obtained a warning for all our 
future labors. 






FXiUATJTV IN OPI'OSITE VAIMATIONS 243 

It is pleasing to notice that in the emph)ynient of the geo- 
metric mean already made in Chapter IV. we have nothing to 
revise ; for we there dealt Avith the geometric' mean proper, 
that is, with two figures — two periods — regarded . 4 equally 
important. 

And we may continue for the present to treat ])rin(!ipally of 
means. 

§ 7. Now in our own special subject people have generally 
confined their attention to price variations, and these price varia- 
tions they have, of course, measured in the usual way, conceiv- 
ing of them all as starting from the same figure. They have 
also generally used the arithmetic average, assuming that the 
proper equality in compensatory changes is the arithmetic. The 
adoption of the arithmetic average we have seen to be due prin- 
cipally to its convenience. But when people have looked for a 
reason to justify its use, they have lighted upon the fact that by 
it alone — measured in the usual way — does equality, especially 
the equality of distance traversed, exist in the opposite varia- 
tions that are supposed to compensate for each other. We now 
see that it is only because of the method usually adopted for 
reckoning percentages that equality is apparent only in the arith- 
metic compensations ; and yet this is only one of three possible 
methods. The usual method is the most convenient and the 
most suitable for measuring the proportions of the variations ; 
but it is no better than either of the others for measuring the 
actual amounts, or distances traversed, of the variations. We 
therefore find ourselves in need of special reasons in our subject 
for showing why the one of these sets of starting points is the 
proper one rather than another. But for this purpose we must 
ap})eal to facts ; for only facts will show what the starting points 
really are. Facts will give us the weighting, and the period, or 
periods, of the weighting. And the mean, or average, will then 
have to be chosen with dependence upon the period, or periods, 
whose weighting is chosen. Otherwise our procedure will be 
arbitrary. 

In general these principles, derived from what has above been 
shown at the end of § 4, may be stated : — (1) If it happens that 



'244 GENERAL ARGUMENT FOR THE GEOMETRIC MEAN 

the things that vary are equal at the start and diverge, using 
even weighting, we should use the arithmetic mean. (2) If it 
happens that the things are unequal at the start and converging 
become equal at the finish, using even weighting, we should use 
the harmonic mean. (3) If it happens that the things are un- 
equal at the start and at the end are reversely unequal, having 
exactly changed places, then, using even Aveighting, we should 
use the geometric mean. Or, putting the matter still more 
broadly, whatever the weighting be, provided we are dealing 
with the same material things at both periods, (1) if we use the 
weighting of the first period we should use the arithmetic aver- 
age ; (2) if we use the weighting of the second period, we should 
use the harmonic average ; (3) if we use the weighting of both 
periods, we should use the geometric average, although in this 
case only approximation to a correct result may be expected. A 
peculiar relation exists between these three means (or averages) 
so used (applied to the same things at both periods), which it is 
not the place here to point out, but which will be shown in a 
later Chapter. Some of these jjrinciples are applicable some- 
times also to classes that are not composed at both the periods of 
the same things. The examination of these cases, so important 
in our subject, must likewise be postponed. 

Now if we simply suppose we are dealing with equally large^ 
or equally important, things, or classes, the natural implication 
is that they are equally large or important, not at only one of 
the periods, but either at both, or somehow over both together. 
Therefore in such general suppositional cases the proper mean or 
average to use with even weighting is the geometric. 

§ 8. Furthermore, in our subject we should remember that 
we have to look at the variations in two aspects. Now if a 
person finds to his own satisfaction that the arithmetic average, 
with a certain weighting, is the right one for the averaging of 
price variations, because, perhaps, of the existence of equality of* 
distance traversed in these variations when supposed to be com- 
pensatory, he will have to use, with the same weighting, the 
harmonic average for the variations of the exchf age- values of 
money, or of the mass-quantities purchased, in spite of the fact 



NATURES OF THE SUBJECTS 245 

that the eciualitv of distance traversed is now lost (although, in 
certain cases, as we shall see, he may still use the arithmetic 
average, but with different weighting). And reversely, if he 
prefers the arithmetic average for averaging the variations of 
the exchange-values of money, he will have to use, with the 
same weighting, the harmonic; average in averaging the price 
variations, in spite of the loss here of equality of distance trav- 
ersed (although, in certain cases, as before, he may still use the 
arithmetic average, but with different weighting). Hence each 
of these ])ersons ought to be called upon to give other reason 
for his position than the mere preference for arithmetic equality, 
since he can have this (except in certain cases) only in one of 
the two aspects of the problem. He should state also why the 
harmonic average is to be preferred on the other side of the 
problem. It is only the advocate of the geometric average who 
can use the same average, with the same Meighting, and the 
same arguments for it, on both sides of the problem. And if 
his arguments should suffer by comparison with the arguments 
for the arithmetic average, they might gain by comparison with 
the arguments, or lack of arguments, for the harmonic average, 
which the others are also called upon to advance. As no argu- 
ments have ever been adduced for the harmonic average directly, 
we shall generally have to compare the arguments for the geo- 
metric average only with the arguments for the arithmetic 
average. 

II. 

§ 1 . We shall now turn our attention to the nature of the 
subjects to which the different averages may be applicable. 

In some subjects the zero point is a purely arbitrary position, 
or if not arbitrary, it is at least a point that can be passed. 
When a positive figure ])a'sses such a zero, it is to be treated as 
negative ; and a negative figure, passing it, becomes positive. 
A familiar example is the figure representing temperature in 
our usual methods of measuring temperature, which figure may 
be above or iiclow zero. In all these cases there is no reason 
apparent why compensation by arithmetically equal variations 



246 GEXERAT. ARGUMENT FOR THE GEOMETRIC ME AX 

should not be the proper one ; and m fact we generally find it 
to be so. The followhag is an example. On a mathematical 
balance two equal weights are in equilibrium at equal distances 
on each side of the fulcrum, which point is represented by zero, 
the one weight being at the distance, say + a, from it on the 
riffht, and the other at the distance — a from it on the left. 
Xow if the weight on the right be split into two equal parts 
and the one of these be moved further to the right, the other, 
in order to keep the equilibrium, must be moved to the left to- 
ward the fulcrum at an equal speed, the two always being at 
equal distances from their original position, whose distance from 
the fulcrum, + a, is thus the arithmetic mean between their dis- 
tances from the fulcrum. This is possible because when the 
part moving to the right passes the distance + 2 a, the other part 
moving to the left may pass the zero point. Thus, for instance, 
when the former is at + 3 a, the latter will be at — a, and these 
are the arithmetic extremes around -)- a J 

Again there are subjects in which, while the one thing, in 
falling in some attribute, cannot pass the zero point, the other 
thing in rismg likewise can not pass a definite point equally far 
above the starting point (supposing even weighting to be proper). 
For instance, if two parts of the territory of a country are 
equally populous and the population of the one part gradually 
moves over to the other part, as the population of the former 
decreases toward zero, the population of the latter rises toward 
double, — the total being supposed to remain the same ; — and so, 
the population of each half being represented as 1 at the start, 
as the one cannot sink below 1 beyond 0, the other cannot rise 
above 1 beyond 2. In this movement, the population of the 
whole country remaining the same, the average population of its 
parts remains the same. This constancy is observed by drawing 

^ Here we are using even weighting because the two parts are equal. If the 
one were half the other, i. e., if the original weight were divided into three equal 
parts and two of them move together, the latter would have to move only half as 
fast as the former. For instance, + a is the arithmetic mean between + 3 a twice 
repeated and —3a; and when the larger part has moved from + a to +3 a, 
traversing the distance +3a~a = 2a (wherefore each of its halves traversing 
this distance, the two halves together have moved over +4 a), the smaller part 
(equal to each of the halves of the other) has passed from + a to — 3 a, travers- 
ing the distance — 3a — a = — 4a. 



NATURES OF THE SUBJECTS 247 

the arithmetic average of the variations, with even weighting.^ 
And the compensation is by arithmetic equality, with equality 
of distiince traversed away from the conmion starting j)()int. 

Reversely there are subjects in which a common point may be 
approached, on opposite sides, from points within two impassable 
limits, the one of these being zero and the other some definite 
figure. Thus in the preceding example reversed, if the popula- 
tions in the two parts are imequal at first and by the moving of 
people from the larger to the smaller become equal, then, the 
total population remaining unchanged, the constancy of the 
average of the two parts will be found to be indicated by the 
harmonic average, with even weighting.^ Here the compensa- 
tion is by harmonic equality, or equality of distance traversed 
to the common ending points. 

But there are subjects in which the zero point is absolute and 
imjjassable, a negative quantity having no meaning, or there 
being nothing that can be conceived of as negative, while on the 
rising side there is no fixed limit. In these the use of the arith- 
metic average for measuring compensatory variations is obviously 
out of place, because the conditions here permit the existence of 
variations on the one side which cannot be arithmetically com- 
pensated on the other. It is evident that a variation moving in 
an unlimited direction is not properly compensated by a varia- 
tion moving at the same speed toward a fixed limit. The arith- 
metic average is admissible only in the two cases just reviewed : 
— either where there are no limits on either side, or where there 
are fixed limits on both sides. Then the two limits should be 
reached at the same moment by the outward moving quantities. 
Now when zero is a limit on the one side and on the other there 
is no finite limit, we can conceive of infinity as being the limit 
on this side. Compensatory variations would then be such that 

2 Even weighting is to l)e employed beeause of the equality of the population 
in the two parts at the first period. If the one part were larger than the other at 
the first period, its weight would have to be proportionally larger. 

' Even weighting is employed because the populations of the parts are equal at 
the second period. If they finish with different sizes, uneven weighting accord- 
ing to these must be employed. — The mathematical reader will perceive an 
agreement, in such cases, between the the two different averages with the two 
different weightings. This is part of the peculiarity above alluded to, which will 
be examined later (in Chapter X.). 



248 GENEEAL ARGUMENT FOR THE GEOMETRIC MEAN 

the one approaches infinity at the same rate as the other ap- 
proaches zero. Such are geometric variations. In these the 
falling quantity reaches zero no sooner than the other reaches 
infinity. Hence whenever we have these limits, the geometric 
average is the proper one for measuring compensatory varia- 
tions, or for indicating their combined results. 

§ 2. Now exchange- values, and prices, are exactly such quan- 
tities. They can rise infinitely. They cannot sink below zero 
or nothing. It is true that we sometimes pay to get rid of a 
thing, as a disease, or an obstruction of any sort. Such things 
may be represented as having negative exchange-value, or nega- 
tive price. Also things that have positive exchange-value may 
sink to negative exchange-value either through deterioration or 
through superfluity. But there are few instances of things that 
have positive exchange-value sinking to negative exchange-value 
when new. And even if they were common, nobody would claim 
that, [A] and [B] being two equally useful classes of things diffi- 
cult of attainment, il [A] should more than double in price, this 
could be compensated by [B] becoming a repugnant object difficult 
of avoidance. Instead of remaining in a constant position with 
respect to these two objects together, we should be doubly in- 
commoded, as we should have to pay more to procure the one and 
additionally pay to get rid of the other. And when a class of 
things, through a change in our wants, falls to zero in price or 
exchange-value, it is no longer produced, it becomes an extinct 
class, and is no more to be taken into account in our comparisons. 

Therefore no rise of A in price from 1 .00 to a height above 
2.00 can be arithmetically compensated by any fall of B from 
1 .00, if they belong to equally important classes, since B would 
have to fall below zero, which here is meaningless. Hence in 
these cases there can be no arithmetic compensation. 

As regards the harmonic compensation, the trouble is inverted. 
We have seen that in this kind of compensation, if B falls in 
price from 1.50 to 1.00, A should rise in price from .50 to 1.00. 
Then if B falls in price from 2.00 to 1.00, A should rise in 
price from to 1.00. And if B falls in price from any figure 
above 2.00 to 1.00, that is, if it falls by more than half, A 



APPLIED TO EXCHANGE-VALUES 249 

should rise in price from below zero to 1.00, whieh i,s impos- 
sible. Hence in such cases there can be no harmonic 
compensation. 

Similarly from the ])oint of view from wliieh oui* problem 
must be regarded, if the exchange-value of money in [B] rises 
by more than 100 per cent., the eompensatijry fall of the ex- 
change-value of money in [A] should be to below zero accord- 
ing to the arithmetic method of averaging, which therefore is in- 
applieahle in this case. And if the exchange-value of money in 
[A] falls to less than half, the exchange-value of money in [B] 
should rise from below zero according to the harmonic method 
of averaging, which therefore is inapplicable here. 

But in the use of the geometric compensation there are no 
such impossible cases. If the price of A rises from any figure 
to any figure, we merely have to suppose that the price of B falls 
from the latter to the former figure, or in the same proportion. 
And if the price of B falls from any figui ^ to any figure, we 
merely have to suppose that the })rice of A rises from the latter 
to the former figure, or in tlie same proportion. Similarly, 
though inversely, in the case of the exchange- values of money 
in the two classes. There is no conceivable case in which this 
kind of compensation is not possible — that is, on the supposition 
we are all along making, in our problem about exchange-values 
or prices, that the classes are equally large or important at, or 
over, both the periods together. 

III. 

§ 1 . Having examined the general nature of the three means 
and averages and of the subjects to which they are applicable, 
we may now examine the means and averages as applicable to 
our special subject. 

The advocate of the arithmetic averaging of prices says that 
when the class [A] rises in price by a certain percentage, in 
order to compensate for this rise and to keep the avernge of 
prices constant, the class [B] should fall by an equal percentage. 
Usually the writers on this subject have not much concerned 
themselves about weighting, but when their attention is turned 



250 GENERAL ARGUMENT FOR THE GEOMETRIC MEAN 

toward it, they posit merely that the classes in question should be 
equally important. They have not entered into the question at 
what period the classes should be equally important. We may, 
then, at present, take our own position on this subject, and re- 
quire that the classes be equally important, or large, over both 
the periods together — either alternately, or constantly so, or on 
the average. 

We may suppose that as A rises in price by one per cent, at 
a time (always reckoned on the original starting point at 1.00), 
B falls by one per cent, at a time (likewise always reckoned on 
the same original starting pomt at 1.00). According to the 
arithmetic averagist these opposite movements always leave the 
exchange-value of money and the level of prices unchanged — 
until 2.00 is reached by A, after which he does not concern 
himself further (or requires that another equally important class 
shall begin to fall). But, now, every successive rise of A is a 
smaller rise reckoned from its starting point, while every suc- 
cessive fall of B, similarly reckoned, is a larger fall. Thus, 
while the first rise and fall are each by 1 per cent., the second 

/1.02 - 1.01\ 
rise of A from 1.01 to 1.02 is a rise by 100 I j-^ j 

= 0.990099 per cent., and the corresponding fall of B from .99 

/ 99 _ 98\ 
to .98 is a fall by 100 i' 99^—)= 1-0101 per cent. The dif- 
ference will be shown more plainly by an extreme case. When 
A has risen to 1.98 and B fallen to .02, the next rise of A to 

/l 99 _ 1 98\ 
1.99 is a rise by 100 ( ' j= 0.505 per cent. ; but the 

/.02-.01\ 
next fall of B to .01 is a fall by 100 ( ^ — j= 50 per cent. 

Now suppose we begin with things in this condition. The price 
of A being 1.98, the price of 0.505 A is 1.00, and the price of B 
being .02, the price of 50 B is 1.00. We may, then, form a new 
unit for each of these quantities, say A' priced at 1.00 and B' 
priced at 1.00 — and we may still suppose that the classes [A] 
and [B] are equally important. Then the arithmetic averagist, 
if he says the rise of the price of A to 1.99 is compensated by a 



Al'PLTED TO EXOITAN(JE-VAIAIES 251 

fall of the price of B to .01, must sav tluit the rise of the price 
of A' to 1 .00505 is compensatecl by the full of the price of B' 
to .50 ; which is absurd on his own principles. Thus the posi- 
tion of the arithmetic averagist leads him into inconsistency. 
And on ji;oing back to the start, we now see that even the first 
fall from 1.00 to .99 is a larger fall than the rise from 1.00 to 
1.01, if we analyze each of these variations into component 
stages. Therefore all such price variations mean an apprecia- 
tion of money. 

On the other hand, in harmonic variations the fall demanded 
in compensation for a rise is always smaller than the rise. This 
is so obvious that it does not need explication. But the position 
of the harmonic averagist of prices may be shown to be wrong 
also in this way. His position is that we should have arithmetic 
equality in the compensatory changes in the quantities of things 
purchasable, whereby the exchange-value of money in their 
classes is measured. Here of course the same absurdity comes 
to light, with the c(jnsequent inconsistency, as in the case where 
the arithmetic average is applied to prices. And the falls in 
the exchange- values of money being too large, this kind of varia- 
tions means a depreciation of money. 

§ 2. The geometric method of averaging escapes such absurd- 
ity and such inconsistency. It })rovides, not equality of rate of 
variation (measured in the usual way), but correspondence. As 
the rate of the rise of A, supjjosed to be always at the same speed 
from the original starting point, grows smaller with every ad- 
vance, so the rate of the fall of B is made to grow proportion- 
ally smaller. Thus when A rises from 1.00 to 1.01 by 1 per 

cent., the compensatory fall of B is from 1.00 to ^ ^ = .990099, 

Avhich is a fell by 100(1 - .990099) = 0.990099 per cent. 
Then when A next rises to 1.02, the compensatory fall of B is to 

- - ^^ = .980392. Here the percentage of the rise of A, reck- 

1 . . . . , , ^/ 1-02-1.01 \ 
oned m its startnig ponit, has sunk to 100 ( :j-^ | = 

0.990099, and the percentage of the fall of B, reckoned in 



252^ GENERAL ARGUMENT FOR THE GEOMETRIC MEAN 

^^^/.990099-. 980392 \ 
its starting point, has snuk to 1001 oonnQQ )~ 

0.980407. Again when A next rises to 1.03, the compensatory 

fall of B is to —TTi, = .970873. Here the percentage of the 

rise of A, reckoned in its starting point, has sunk to 

/l.()3 _ i.02\ 
100 ( —^1)9 ) ^ 0-980392 (the same as the point to which 

B previously fell), and the percentage of the fall of B, reckoned 
in the same way, has sunk to 0.970938 per cent. And so the 
process will continue indefinitely, the percentage of the fall of 
A always being the same with the point to which B previously 
fell, and the percentage of the fall of B always being still smaller 
(though above the point to which B falls). For instance, in the 
ninety-ninth stage, the rise of A from 1.98 to 1.99 is compensated 

by the fall of B from .^- = .505050 to ^-Kq = -502512, so 

/1. 99 — 1.98\ 

that the rise bv 100 ( ^ '- — ) = 0.505050 per cent, is com- 

V 1.98 J ^ 

/ 505050 502512\ 

pensated by a fall by 100 (^^ ' ' -^ J = 0.502524 

per cent. 

Now the first compensatory fall of B from 1.00 to .990099 is 
the same variation as a fall from 1.01 to 1.00, — merely the re- 
verse of the rise of A for which it is offered m compensation. 
And the next fall of B from .990099 to .980392 is the same 
variation as a fall from 102 to 101, — again merely the reverse 
of the rise of A for which it is offered in compensation. And 
so with all the compensatory variations required by the geometric 
averaging (with even Aveighting). They are merely the reverse 
of each other. Whenever A rises from any figure, a^, to any 
figure, a^, by a percentage (reckoned in a^) obtained by dividing 
a hundred times the difference which measures the amount of 

the rise by the earlier figure, namely 100 I ~ ^ ) , the com- 
pensatory fall of B is a fall from a^ to a^ by a percentage (reck- 
oned m ^2) obtained by dividing a hundred times the differenc^e 



AIM'LIKU TO EXCIIAX(iK-VALT^I-:S 253 

which lucasiires the amount of" the fall by the earlier figurCj 

namely 100 ( -\ . These percentages are not the same, 

but it is evident that this fall is e([ual to that rise. If A rises 
from rtj to a., and then falls back to a^, it is evident that the fall 
is equal to the rise, since it brings A back to its original ])osi- 
tiou. Then the fall of B from a^ to ttj, equalling the fall of 
A from a.^ to a^, equals the rise of A from a^ to a^.^ The 
usual method of reckoning percentages does not manifest this 
e([uality, which is shown mathematically only by reckoning 
the percentages from the opposite extremes, but in the same 
direction. That the mathematical equality should reside only 
in this kind of j)ercentage is plain, since the compensatory 
variations must plainly be merely the reverse of each other. 
Therefore the correctness of the geometric mean (although not 
of the geometric average) is demonstrative. 

§ 3. Jevons wrote in his Prineiples of Science : ^ — " In almost 
all the calculations of statistics and commerce the geometric 
mean ought, strictly speaking, to be used. If a commodity rises 
in price 100 per cent, and another remains unaltered, the mean 
rise of a price is not 50 per cent, because the ratio 150 : 200 is 
not the same as 100 : 150. The mean ratio is as unity to 
l/l.OO X 2.00 or 1 to 1.41 ." There is exaggeration in the first 
part of this statement, since the geometric mean is almost ex- 
clusively to be confined to the measurement of variations, and 
many calculations of statistics are not measurements of variations 
— though often providing the data for such measurements. In 
this statement we have our problem solved in the first form in 
which we approached it. Therefore we may find interest in 
noticing this also. As Jevons's brief statement has not met with 
acceptance on the part of statisticians of prices, it needs to be 
explicated. 

The arithmetic averagist would say that the mean rise in the 
above suppositional case is to 1.50. Now suppose, the price of 
B remaining at 1.00, the price of A rises first of all to 1.50. 

^ Cf. the reasoning for Proposition XI. 
2 Second ed., p. 3G1. 



254 GENERAL ARGUMENT FOR THE GEOMETRIC MEAN 

Then the arithmetic averagist would say the mean price has 
risen to 1.25, Suppose that later the price of A rises from 1.50 
to 2.00. This rise, measured from its starting point, is a rise 
by 33| per cent. Then, according to the principles of the arith- 
metic averagist, the mean, already risen to 1.25, ought to rise 
further by half of 33^, or 16f, per cent, above 1.25. This is 

a rise to 1.25 x 1.16f = 1.45433 Therefore on his own 

principles the arithmetic averagist is mistaken in saying that 
when A rises to 2.00 the mean rise is to 1.50. And conse- 
quently, too, he was mistaken when he said it rose first to 1.25, 
and his whole position from beginning to end is inconsistent 
with itself, and wrong. The mean price at every rise is below 
the arithmetic mean. 

On the other hand, the mean price is always above the har- 
monic mean, because if it rose only to the harmonic mean, the 
mean exchange- value of the two things would fall to 'the arith- 
metic mean, and a similar inconsistency would be found. 

The error in the position of the arithmetic averagist is evi- 
dent. The higher A rises from its original position, the smaller 
is its rise in each stage of its advance. Yet the arithmetic aver- 
agist accords to it the same influence upon the mean when it 
rises from 1.99 to 2.00 as when it rose from 1.00 to 1.01, al- 
though the mean has lagged behind somewhere below 1.50, 
where its rise by half of one per cent, at a time (reckoned in 
1.00) is considerably more than half as large as the rise of A 
from 1.99 to 2.00. And reversely, when A alone falls, while 
it is falling from .02 to .01, the arithmetic averagist accords to 
it not so much influence upon the mean as when it fell from 
1.00 to .99, although its fall is now fifty times greater than it 
was then. 

But the geometric averagist, in placing the mean at 1.41 
when A alone rises from 1.00 to 2.00, places it where the in- 
fluence of the later rises is the same as that of the earlier, when 
in the same percentage. Thus if A alone rises first to 1.41 by 
41 per cent., the mean is placed at 1.19, indicating a rise by 
19 per cent. Then when A rises further to 2.00, this also be- 
ing a rise by 41 per cent., the same influence as before is attrib- 



appt.ii:d to exchange-values 255 

utod to it ; for the rise of the mean from l.li) to 1.41 is by 19 
per cent. 

§ 4. Here we might panse, onr labor done, bnt for tlie fact, 
above shown, that what is trnc of the geometric mean is not trne 
of tlie geometric average. All that has just been demonstrated 
has been demonstrated on the supposition that we are dealing 
with only two classes of things, and these equally large or im- 
portant over both the periods compared. If the two classes are 
not equally large over both the periods, or if there are many 
varying classes to be considered, nothing that has just been 
proved applies, — nor does it apply with accuracy if we use the 
geometric average with the proper weighting over both the 
periods. For instance, suppose the classes are equally impor- 
tant only at tlie first period, and as [A] rises in price it rises 
also in importance, and as [B] falls in price it falls also in im- 
portance. Then it is evident that a rise of A from 1.00 to 1.50 
is not fully compensated by the fall of B to the other geometric 
term, 0.6(3f ; but, for compensation, B must fall further. How 
ranch further it must fall will depend upon the extent of the al- 
teration i^ the relative importance of the classes. It is possible, 
therefore, that the price of B may have to fall to the arithmetic 
term, 0.50. We shall in fact find this to be the case when the 
relative sizes of the classes vary exactly as their prices. Yet 
this will not be exactly indicated by the geometric average with 
the proper (uneven) weighting for both the periods. 

Therefore we need to turn to the examination of the averages 
more closely in connection with the subject of weighting. Also 
it is always well to examine the arguments of persons who have 
advocated other opinions, and although we have already dis- 
proved the positions of the arithmetic and harmonic averagists 
under certain provisos, yet their general position still remains. 
It happens that both these objects may be pursued together. 



CHAPTER IX. 

REVIEW OF THE ARQUMEISTTS FOR THE HARMONIC AND 
ARITHMETIC AVERAGES OF PRICE VARIATIONS. 



§ 1 . The arguments for the harmonic and arithmetic averages 
have been very imperfectly stated. We shall therefore have to 
try to understand them not merely as they have been presented, 
but as they turn out to be on fuller analysis. They have gen- 
erally been adduced in the form which considers what constitutes 
variation. But this depends upon what constitutes constancy. 
Also little or no reference has been made to weighting. This 
we shall have to supply. We may, however, supply it later, 
first reviewing the arguments in their most general forms. 

In the argument for the use of the harmonic average of price 
variations the idea, when there is supposed to be constancy, is 
as follows. The same total sum of money purchasing the same 
total quantity of all kinds of goods at both the periods com- 
pared, the purchasing power of this sum (and consequently the 
exchange-value of money) is considered not to have varied, what- 
ever be the changes in the make-up of the total quantity of 
goods, an increase in the quantity of one class of things pur- 
chasable with the same particular sum of money devoted to pur- 
chasing it at both the periods being offset by an equal decrease 
in the quantity of another class of things purchasable with the 
particular sum devoted to purchasing it at the two periods. 
For the total of the quantities remains the same when these 
quantities vary oppositely to the arithmetic terms, and conse- 
quently when their prices vary oppositely to the harmonic terms ; 
so that if the harmonic average of the price variations indicates 

256 



ANALYSIS OF 'I'lnO AHCMMEN'I'S 257 

constaney, it indicates this conditioii. A.s the (juantities of the 
goods arc in arithmetic progression, the compensation may be 
described as arithnietic compciisafioa hi/ equal maaa-qnantities — 
a loss of one third on A, for instance, being comi)ensated by a 
gain of one third on 1>, or by a gain of one sixth both on B and 
on C Now if a variation occurs in the t(jtal (luantity of the goods 
purchasable at the second period from that at the first, this 
variation of the total is indicated by the arithmetic average of 
the variations of the particular quantities; for if the (piantities 
purchasable all varied at the same common average rate, the 
same result in the total would be obtained. Thus if there is a 
loss of one third in the (luantity of A alone, this is the same, 
among two classes of commodities, as a loss of one sixth on each ; 
among three classes, of one ninth on each ; and so on, just the 
same as if each of the two classes had lost one sixth, or each of 
the three classes one ninth, and so on/ Hence this same result is 
obtained, for the prices, l)y the harmonic average of the price vari- 
ations. All this is a mathematical fact. Upon this fact is based 
the argument, if argument it may be called, or rather the claim, 
suggested by Jevons and insisted upon by Messedaglia, that the 
harmonic average of the price variations, because it inversely 
indicates this variation, or the preceding constancy, of the total 
quantity of goods purchasable with the same total sum of money, 
also inversely indicates the variation, or constancy, of the punjhas- 
ing power of money (and consequently of its exchange-value), 
and so is to be taken as the proper method of measuring it. 

In this argument it is always miderstood that the ])articular 
sums of money spent on every class of goods remains constant 
at both the pei'iods, whatever be the variations in the particular 
mass-quantities therewith purchased. 

§ 2. In the argument for the use of the arithmetic average of 
price variations the idea, when there is supposed to be constancy, 
is as follows. The same total quantity of goods being purchas- 
able with the same total sum of money at both the periods, a 
total price is conceived of the total quantity, which total price 

' This will be recognized as the argument we lighted upon at tirst glance above 
in Chapter VI. Sect. I. §2. 

17 



258 THP] ARGUMENTS FOR THE OTHER AVERAGES 

lias not varied whatever be the changes in its make-up, that is, 
in the particuhir prices, an increase in the price of one class of 
things being offset by an equal decrease in the price of another. 
For the total of the prices remains the same when they vary 
oppositely to the arithmetic terms (and consequently w^hen the 
mass-quantities purchasable vary oppositely to the harmonic 
terms) ; so that if the arithmetic average of the price variations 
indicates constancy, it indicates this condition. And so there is 
arithmetic corapensation by equal sums of money — a need, for in- 
stance, for one half more money to purchase A being offset by a 
need for one half less money to purchase B. Now if a varia- 
tion occurs in the total price, or total simi of money needed to 
purchase the same quantities of the same classes of goods at the 
second period from what was needed at the first, this variation 
of the total is indicated by the arithmetic average of the partic- 
ular prices ; for if the prices all varied at the same common 
average rate, the same result in the total price would be ob- 
tained. Thus a rise in the price of one class of commodities by 
fifty per cent, has the same influence upon the total as a rise of 
two classes by t^veuty five per cent., of three classes by sixteen 
and two thirds per cent., and so on. Hence the result in the 
variation of the total price is obtained by the arithmetic average 
of the variations of the particular prices. Here again is a 
mathematical fact, upon which is based the argument, or claim, 
that the arithmetic average of the price variations, because it 
directly mdicates this variation, or the preceding constancy, of 
the total price of the same goods, also inversely indicates the 
variation, or constancy, in the purchasing power of money (and 
consequently in its exchange-value) and so is to be taken as the 
proper method of measuring it. 

In this argument it is always understood that the particular 
quantities purchased of every class remain constant at both the 
periods, whatever be the variations in the particular sums of 
money needed to purchase them. 

§ 3. Thus the harmonic and arithmetic averages of prices, in 
the minds of their advocates, represent reversed positions. The 
former is directly a])]>lied to the measurement of the purchasing 



ANAL^'SIS OF rilK AIMJl'MKNTS 259 

j)()\\('r of money by the total (|uaiitity of all coniiuoditics a given 
total stun of money, spent in the same way at both |)(!rio(ls, will 
purchase. The latter is applied rather to the measnrement of 
the power of all commodities over money by the total sum of 
money a total quantity of given particular conniiodities, composed 
in the same way at both periods, will command in exchange. 
The former makes use of arithmetic comj)ensation bv e(pial mass- 
cpiantities, the latter of arithmetic com|)cnsation by e([ual sums 
of money. They have in common the use of arithmetic com- 
pensation. They differ in applying this to opj)osite sides of the 
(piestion, in each case excluding notice of the other side. 

When we look at either of these positions by itself, it seems 
very strong. The one inversely measures variations in the total 
mass-quantity purchasable with a given sum of monev. Does 
it not then inversely measure variations in the purchasing power 
or exchange-value of money, and directly measure variations in 
the general level of prices ? The other directly measures varia- 
tions in the total price of given quantities of all things together. 
Does it not then directly measure variations in the general level 
of prices, and inversely variations in the ])urchasing power oi- 
exchange-value of money ? 

Each of these methods is founded on a procedure which w(> 
employ with regard to single classes of commodities. We 
measure the constancy or variation of the })articular exchange- 
value of money in any one class of commodities by the constancy 
or variation of the quantity of this commodity jnirchasable with 
a given sum of money at each of the periods (according to Proj)- 
osition L), — and by inverting the result so obtained we can 
measure the constancy or variation of the price of this thing (al- 
though we never adopt this roundabout course). Hereupon the 
harmonic averagist of prices concludes that we can measure the 
constancy or variation of the general exchange-value of money 
in all things by the constancy or variation of the total quantity 
of things purchasable at each of the periods with a given total 
sum of money (which he further specifies must be spent in the 
same way at both the periods), — and the inverse of the result so 
obtained he regards as the proper measure of the constancy or 



260 THE AEGUMEXTS FOR THE OTHER AVERAGES 

variation of prices in general. Again we measure the constancy 
or variation of the particular price of any one class of commod- 
ities by the constancy or variation of the price of a given quan- 
tity of this commodity at each of the periods, — and the inverse 
of this gives the constancy or variation of the particular ex- 
change-value of money in that class (according to Proposition 
X.). Hereupon the arithmetic averagist of prices concludes 
that we can measure the constancy or variation of a total price 
of all things by the constancy or variation of the total price at 
each period of a given total quantity of all things (further speci- 
fying that these quantities must be individually the same at 
both periods), — and the inverse of the result so obtained (the in- 
verse of the constancy or variation of this total price of all the 
same things) he regards as the proper measure of the constancy 
or variation of the general exchange-value of money in all things. 

In the two particular measurements that serve as models for 
these two general measurements there is no disagreement pos- 
sible. The results obtained by the one are universally the same 
as the results obtained by the other. Those two particular meas- 
urements are also always applicable to every one and the same 
case. The two copies, however, are really applicable each to a 
diiferent state of things, so that they are not properly contra- 
dictory or antagonistic. Yet this fact has been mostly over- 
looked, and the advocates of these different averages have simply 
urged the use of the one or of the other method for all cases. 
When applied to the same cases, differently garbling them to fit 
them to the different requirements, these methods give different 
results. This disagreement shows that at least one of them is 
false, and probably both, when applied in such a loose way. 
But confined each to the cases that may happen to exist to which 
it is applicable, they may both be true, or they may both be false, 
or the one may be true and the other false. At all events this 
divergence of the copies from the models shows that something 
is wrong in the copying at least in one case. 

§ 4. The faultiness of the copy in the position of the harmonic 
averagist of prices is not far to seek. In measuring the con- 
stancy or variation of the particular exchange-value or purchas- 



A.^Al,^■sIs OF TiiK AHcrMFA'i's 2(n 

ing j)()\v{'r of inoiicy in any class of commodities by the (H)nstaucy 
or variation in tiic (quantity of it purchasable at each period with 
a given sum of money, we are careful to note that the (juality of 
tliis inass-(piantity must not change, and guarding this, there is 
no possibility of divei'genee in our result, no matter what mass- 
unit w(! use. Now when wa try to imitate this oj)erati()n in 
measuring the constancy or variaticm in the general exchange- 
value oi' j)ur('luising power of money in many or all classes of 
commodities, we at once strike upon a difficulty if there is the 
least change in the relative exchange-values or prices or pre- 
ciousness of these classes. For if there is any such change, 
this is the same as a change in the (luality of the total mass- 
(piantity, which renders the comparison of its change of size 
nugatory, unless it can be allowed for. Furthermore, the com- 
parison of the total mass-quantities purchasable, or actually pur- 
chased, at each period \v\t\\ the same sums of money, will now 
be different according to the mass-units that are used in each 
class. The harmonii; averagist of prices adopts the usual prac- 
tice in these matters and takes as his mass-unit in every class the 
mass that is equivalent to the money-unit at the first period. 
But this is only one of many possible ways of selecting the 
mass-units, and he has offered no reason for adopting it — and 
a])])arently has none, except the blind following of a convenient 
habit. Doing so, when he finds the mass-quantities purchasable 
with the same sums at each period to foot up to the same total 
quantity, so that he concludes that the exchange- value of money 
has not altered on the whole, it may be that the total weight, or 
the total bidk, of the goods so purchasable with the same sums 
of money may be very different at the two periods. Hence he 
is here not following his model. And if he should now try to 
return to his model, he would be confronted with the (piestion. 
Is he to require the same total weight, or the same total bulk ? 
Between these two there is nothing to decide. But if he does 
(arbitrarily) hit upon the one or the other, he will only be meas- 
uring the constancy or variation in the one or in the other kind 
of preciousness of the goods relatively to money. ^ Thus the 
- He WDuld he using Drobisch's method applied to cases in which x^ai =.1-202, 



262 THE ARCiUMENTS FOR THE OTHER AVERAGES 

measurement of the general exchange-value of money cannot 
be so simply made on the model of this method of measuring a 
particular exchange-value of money. The imitation will be ex- 
act, in case the total mass-quantities are the same at both periods, 
only if there are no price variations, and, in case the total mass- 
quantities are different, only if all the prices have varied in the 
same proportion — that is, only when any other average would 
be as good. There is, however, a more complex way in which this 
model may be imitated. But the harmonic averagist has not 
sought it, and nobody has hitherto pointed it out. 

In the argument for the arithmetic average of prices there 
is a somewhat similar defect. But this seems to bear with it its 
own correction, in the way alluded to at the end of the preced- 
ing Chapter. Here, m its totality and in its details, identically 
the same (or similar) mass of goods is used, of which the con- 
stancy or the variation in the total price is taken as the measure 
of the constancy or inverse variation in the exchange- value of 
money. And consequently there is no difficulty here about the 
mass-units to be employed. But this- total mass is not econom- 
ically the same in all its parts at both periods, unless all prices 
have remained constant or varied alike ; for otherwise some of 
its parts have become more or less precious, and also more or 
less important, than others, and so its economic make-up has 
altered. Still it is precisely the prices of the things that have 
grown more precious and more important that have risen, and 
the prices of the things that have become less precious and less 
important that have fallen, and all these variations are in ex- 
actly the same proportions ; which is about as we should desire. 
Hence it is possible that this measurement is a good copy of its 
model. But it would probably be more difficult to prove this 
than to prove the correctness of the arithmetic average (in the 
special cases to which it is claimed to be applicable). Hence 
the above argument made by the arithmetic averagists is still 
unsatisfactory, and we must continue to probe it. 

2/i/3i =2/2/32> (because the same sums are supposed to be spent ou every 

class at both periods), so that the first part of Drobisch's formula falls away, 
being reduced to unity. 



AFATIIKMATICAL K KLATK ).\"S KiNOlfKD 'H\'.\ 

II. 

§ 1. Both the above argiuueiits have been advanced oidy on 
very cursory inspection and incomplete analysis (if the inathe-. 
matieal relations involved. The writers ae(piainted with them 
seem to have th()U<>;ht that the ar<>;urnent for the harmonic aver- 
age is ])eculiar to that average, a|)])lieable neither t(t the arith- 
metic nor to the geometric, and that the argument for the arith- 
metic average is peculiar to that average, apj)licable neither to 
the harmonic nor to the geometric. And as these two argu- 
ments, attacking our ])roblem fn^n its two opposite sides, appear 
to occupy all the possible positions, it has seemed as if no room 
were left for the geometric average or mean. This has seemed 
to stand out in the cold, with no function to fulfil, and with no 
argument applicable to it. Hence the neglect Avith which it has 
been treated. Yet not much analysis is neech'd to show that 
these views are false. 

The argument for the harmonic average of })rices assnmes a 
certain distribution of our spendings, and then finds c(»mpensa- 
tion arithmetically by equal mass-quantities, so that, when con- 
stancy is indicated, an equal total mass-quantity, though differ- 
ently made up, is purchasable at both periods with the same 
sum spent in the same way at both periods. Now suppose 
]>rices have changed to the arithmetic extremes, namely in our 
example from 1.00 to l.oO and from 1,00 to .50. Then if we 
spend 1.50 on [A] at both periods and .50 on [B] at both 
periods, we get for 2.00 at the first period 1^ A and ^ B, and 
at the second period 1 A and 1 B, — losing | A and gaining | B. 
Thus there is compensation by equal mass-quantities, and ability 
with the same sum to purchase an equal total mass-<iuantity. 

The argument for the arithmetic average of prices assumes a 
certain distribution of our purchases, and then finds compensa- 
tion arithmetically by equal sums, so that, when constancy is 
indicated, the same total sum, though differently made up, is 
able to purchase at both jieriods the same total mass-quantity 
made up of the same particular mass-quantities at both periods. 
Now suppose prices have changed to the harmonic terms, namely 



2(34 THE ARGUMEXTS FOP. THE OTHER AVERAGES 

in our example from 1.00 to 1.50 and from 1.00 to .7o. Then 
if we purchase f A and 1 J A at both periods, we can do so at 
the first period by spending .66f on [A] and 1.33|^ on [B]j 
and at the second period by spending 1.00 on [A] and 1.00 on 
[B] — or 1 M more on [A] and i M less on [B] . Thus there is 
compensation by equal sums, and ability to purchase exactly the 
same particular mass-quantities with the same total sum. 

Suppose, again, that prices have changed to the geometric 
terms, namely from 1.00 to 1.50 and from 1.00 to .66f . Then 
if we spend 1.20 on [A] at both periods and .80 on [B] at 
l^oth periods, we get for 2.00 at the first period Ig- A and | B 
and at the second period ^ A and 1^ B, — losing ^ A and gain- 
ing I B. Thus there is compensation by equal mass-quantities, 
and ability with the same sum to get an equal total mass-quan- 
tity. Also if we purchase ^ A and 1 g^ B at both periods, we 
can do so at the first period by spending .80 on [A] and 1.20 
on [B] and at the second period by spending 1.20 on [A] and 
.80 on [B]— or | M more on [A] and f M less on [B] . Thus 
there is compensation by equal sums, and ability to purchase 
the same particular mass-quantities with the same total sum. 

Hence the argument which seems to be peculiar to the har- 
monic average applies also to the arithmetic and to the geo- 
metric ; and the argument which seems to be peculiar to the 
arithmetic, applies also to the harmonic and to the geometric. 
And the geometric average, or at least the geometric mean, in- 
stead of being without any argument, and without any function, 
equally is subject to either of the arguments, and equally per- 
forms the same functions. Instead of standing out in the cold, 
it belongs in the fold ; and we cannot examine either of the ar- 
guments far without taking also it into consideration. 

Thus are there two arguments, in the forms hitherto used, for 
three means or averages, each argument being found to be appli- 
cable to each of the means. Therefore, so far as we yet see, 
any one mean can apparently be argued for as well as any other 
by either of these arguments.^ 

^ That there is ability to purchase with the same sum at both periods an equal 
total quantity of commodities differently made up [hence with compensation by 



MATIIKMATKAL RKI.ATfONS KJXOIIED 265 

§ 2. Furthermore, when prices ehaiig-c to the harmonic ex- 
tremes and tlie harmonic; average indicates c )nstancy, or when 
prices change to the arithmetic extremes and the harmonic aver- 
age indicates c(»nstan(!y, or wlien prices (ihange to the geometric 
extremes and the geometric mean indicates constancy, althongh it 
is possible in each case with the same snra of money spent in the 
same way at both periods to purchase an equal total (quantity of 
goods differently made np, the spcndings and the purchases be- 
ing different in each of the sup])ose(l changes, it is also [)ossible 
in each ease with the same sum of money spent in the same way 
at both periods to get botli a larger and a smaller total ipiantity of 
goods at the second than at the first period, our spcndings being 
variously distributed. Thus in our simple example of two 
classes of commodities both j)riced at 1.00 at the first period we 
can at that ]K'rio(l itself with 2 M ])urchase 1 A and 1 B, or 
two given quantities of [A] and [B] together, and in any com- 
bination of spcndings of the same sum we always can purchase 
two such tpiantities — e. g., f A and 1^ B, | A and 1| B, 1| A 
and I B, 2 A and B, A and 2 B, etc. But at the second 
period with prices changed to the harmonic terms, 1.50 and .75, 
if we employ 2 M to purchase 1 A and some B, we must use 
H of it in purchasing 1 A and have left only J with which we 
can purchase only f B, or all told o\\\\ 1| of [A] and [B] to- 
gether ; or if we purchase 1 B with f M, with the remaining 1^ 
]\r we can purchase only | A, or 1 1 of [A] and [B] together 
— in both cases a smaller total than before. Or again, employ- 
ing 2 M to purchase \ A and some B, we can purchase If B, 
or together 2^, this time more than before. With the prices 
changed to the arithmetic terms, 1.50 and .50, by employing 
1 M to purchase some A and 1 M to purchase some B, we get 
I A and 2 B, or on the whole more than before ; or by spend- 
ing If M on [A] and \ M on [B] we get l\ A and i B, or 
on the whole less than before. And lastly if the prices change 
to the geometric terms, 1.50 and .66|, we can also get some- 
times more and sometimes less of [A] and [B] together — more, 

equiil mass-quantities] even when the simple harmonic average of the price varia- 
tions indicates constancy, was perceived by Walras, B. G9, pp. 14-15. But Walras 
has not investigated further. 



266 THE ARGUMENTS FOR THE OTHER AVERAGES 

for instance, if we spend 1 M on [A] and 1 M on [B] , getting 
|- A and IJ B, or all told 2|, or less if we spend 1| M on [A] 
and 1 M on [B], getting li A and i B, or all told li|. 

Hence, if the mere possibility of getting, with onr money 
spent in the same way at both periods, an equal total quantity 
of commodities is a reason for thinking our money constant in 
purchasing power, the simultaneous possibility of getting both 
more and less is an equally good reason, so far as is yet shown 
to the contrary, in each case, to think that our money has both 
appreciated and depreciated ; which is absurd. 

Again, although it is possible in each of these cases with the 
same total sum of money differently spent to purchase at both 
periods exactly the same quantities of commodities, these being 
different in each of the supposed cases, it is also possible in each 
case that to purchase other exactly the same quantities of com- 
modities, a larger or smaller sum of money is needed at the sec- 
ond period. This could be easily shown in our simple example. 
But enough has been shown already. 

Hence, if the mere possibility of getting at both periods ex- 
actly the same quantities of commodities with the same total 
sum of money is a reason for thinking the level of prices con- 
stant, the simultaneous possibility that for getting exactly the 
same other quantities of commodities both a larger and a smaller 
total sum may be needed, is equally good reason, so far as is 
yet shown to the contrary, in each case to think that the level 
of prices has both risen and fallen ; which again is absurd. 

Therefore, so far as we yet see, these arguments apparently 
are equally defective whether applied to the harmonic, arith- 
metic or geometric averages or means. 

§ 3. This defectiveness of the arguments seems to have been 
ignored. The conflicting possibilities have been overlooked. 
Hardly any advance has been made in this matter since the 
famous dispute between Jevons and Laspeyres. That dispute, 
therefore, deserves review ; for it may provide a Avarning, still 
needed in our subject, against reasoning which stops at a few 
half-truths first lighted upon. 

Jevons had noticed in his first work that the price of cocoa 



:MATin-;MAri('AL rei^atioxs icxoitKi) 2()7 

had recently risen 100 per cent., while that of cloves had fallen 
50 per cent., and had said it would be "totally erroneous" to 
say the average change was a rise of 25 per cent., since the geo- 
metric mean in this case indicates no variation at all." Here- 
uj)on Las})eyres commented and argued as follows : — " The 
geometric mean expresses neither the depreciation of commodi- 
ties or a])})reciation of money, nor the ap})reciation of commodi- 
ties oi- depreciation of money — that is, according to Jevons, 
increase^ or decrease in its ' })otency in purchasing other articles.' 
Let us retain the example used by Jevons. Here, after the 
change in price of vocxrd and cloves, the same sum of money has 
not the same purchasing power as before, but a smaller one, and 
exactly so much smaller as is indicated by the arithmetic mean. 
If a certain weight of cocoa (say 1 cwt.) previously cost 100 
thalers, and a certain weight of cloves (say 1 cwt.) also cost 100 
thalers, and the ])rice of this amount of cocoa rises from 100 to 
200 thalers, and that of the cloves falls from 100 to 50, then 200 
thalers no longer have the same potency in purchasing cocoa 
and cloves. For this sum the purchaser procures only | cwt. 
cocoa (= 150 th.) and 1 cwt. cloves (= 50 th.), or he procures 
1 cwt. cocoa (= 200 th.) and no cloves at all. The i)un'hasing 
))0Aver is now -i less, that is, the })urchaser must add ^ in order 
to get the same quantity ; or the 250 thalers are now by i (50 
th.) less worth than formerly. Exactly this is expressed by the 

200 -f- 50 
arithmetic mean ^ = 125; 125 thalers have only the 

same purchasing power as 100 l>efore, or 250 only the same as 
200 before. Money has depreciated 20 per cent.; commodities 
have risen 25 per cent. What is true of the average of two 
commodities, is true also for any number of commodities." ^ 

Lasjieyres thus found fault with the geometric average for in- 
dicating constancy in the " potency in purchasing " under the 
given conditions, because these conditions permit us with one 
whole sum, 200 thalers, to get at the second period only 1 cwt. 
cocoa and no cloves, that is, a smaller quantity than before, 

2 B. 22, pp. 23-24. 
» B. 25, p. 97. 



268 arCtUmexts for the other averages 

which fact he took for an indication that the " potency in pnr- 
chasing " was smaller ; to which came the added evidence that 
more money is required at the second period to buy the 1 cwt. 
of each article. He omitted to state that these conditions per- 
mit us at the later period to buy 4 cwts. cloves and no cocoa, that 
is, this tiuie a larger quantity than at first, and one just doubles, 
as the other was half. Had he done so, the mdication of depre- 
ciation would have been no stronger than that of appreciation. 
And he probably failed to see that under these conditions we 
could at the first period purchase with 66.66f thalers f cwt. 
cocoa and with 133.33^ thalers IJ cwts. cloves, or 2 cwts. with 
200 thalers ; and that at the second period we could purchase 
with 133.33^ thalers f cwt. cocoa and with 66.66f thalers 1^ 
cwts. cloves, that is, exactly the same quantities of cocoa and 
cloves, amounting to 2 cwts., with the same total sum of money, 
200 thalers. Had he noticed this, he would have seen that, so 
far as either he or Jeyons had yet carried their investigations, 
the indication of constancy is as strong as the indication of de- 
preciation, his argument oifering nothing distinctive in proof of 
the indication of depreciation made by the arithmetic average 
over against the indication of constancy made by the geo- 
metric. 

And when Jevons in reply suggested the harmonic average 
(in this case .80, indicating a fall of prices by 20 per cent.) on 
the ground that it marks the change in the total quantity of 
commodities the same sums of money will purchase at the two 
periods (here, for 100 thalers spent on each article, 1 cwt. cocoa 
and 1 cwt. cloves at the first period, and at the second J cwt. 
cocoa and 2 cwts. cloves, the arithmetic average being J (J -f 2) 
= 1.25, indicating appreciation of money corresponding to the 
above fall of prices), he probably failed to see that we could at 
the first period purchase with 133. 33J thalers 1^ cwts. cocoa 
and with G6.66f thalers f cwt. cloves, or 2 cwts. with 200 
thalers ; and that at the second period we could purchase with 
133. 33|- thalers f cwt. cocoa and with 66.66f thalers IJ cwts. 
cloves, that is, Avith exactly the same expenditure of 200 thalers 
the same total quantity of 2 cwts., — wherefore even his argu- 



MATHEMATICAL RELATIONS HiNOIJEl) 'H]d 

ment for the harnionic average, so far as he worked it out, would 
indicate constancy as readily as a fall of prices. 

^ 4. We have, even, as yet by no means exhausted the posisi- 
bilities which render such arguments ridiculous. The following 
propositions hold : — The iwloe of at least one class rising and the 
price of at least one class falling, no matter how large or small 
these variations be, it is possible by spending constant suins of 
money on the different classes, with the same total sum to pnircha.ne 
at both periods the same total quantity of goods ; and under the 
same conditions, it is possible at botli periods to purchase consta)it 
quantities of the different classes, consequently the same total quan- 
tity of goods, with the same total sum of money, differently spent 
at the two periods. In the first of these cases the equality in the 
total (quantities of goods at both the periods depends, given the 
price variations, upon two other factors. It depends both on 
the sizes of the mass-units between the numbers of which the 
compensation by arithmetically equal quantities is desired, and 
on the proportions between the special sums devoted at both 
periods to purchasing the different classes. In the second case 
the equality in the total sums of money at both ])eriods depends 
upon only one factor beside the price variations. This one other 
factor is the proportion between the special quantities of the dif- 
ferent classes purchased at the two periods ; but as these quantities 
are affected, in their numerical expressions, by the sizes of the 
mass-units used, these also play a part, though a subordinate 
one, as ^\e shall see. Now this factor, so far as it is a factor, 
of the sizes of the mass-units used, has generally been decided 
at the outset in the same ^vay by the harmonic and by the arith- 
metic averagists. They both employ mass-units that are equiv- 
alent at the first period (being then priced at 1.00 or at 100 
money-units) — as we have just seen done by Jevons and Las- 
peyres. These mass-units being settled upon, the first factor in 
the first case, and so far as it affects the second, is disposed of, 
and now the equality in question depends upon the other factor 
in both cases. In the first case, if an article rises much in price, 
so that the deficiency in the quantity of it purchasable at the 
second period, compared with the quantity purchasable at the 



270 ARGUMENTS FOR THE OTHER AVERAGES 

first, is large, and if another falls slightly in price, so that the 
gain here, quantity for quantity, is small, we only have to ex- 
tend the quantity of this class purchased at the first period until 
its gain at the second period equals the deficiency of the other. 
And similarly in the second case. The number of classes does 
not affect the matter. All but one may rise much and that one 
fall but slightly : still it is possible for the compensation by 
arithmetic equality to take place. 

When we are dealing with only two oppositely varying classes 
it is easy to get formulae, A^^hich may be of service. To begin 
with the first case : — Let the sum of money devoted at both 
periods to purchasing [A] be represented by a, and that de- 
voted at both periods to ]3urchasing [B] be represented by b ; 
let us for the present adopt the usual course and take for our 
mass-unit for each class that mass of it which can be purchased 
at the first period for one money-unit, and let a^^ and \i.^ be the 
prices at the second period of these mass-units of [A] and [B] 
respectively. We therefore purchase with a money-units a 

a 

mass-units of [A] at .the first period and at the second -^ mass- 
units, and with b money-units we purchase b mass-units of [B] 

b 

at the first period and at the second ^, mass-units. Assuming 

that the sums of these mass-quantities are the same at both 

periods, we have 

a b 



which gives us 



^^a,V«_-^). 



<(l-/5/) 
Now let us take a as a unit sum ; then 

which means that for every sum of money spent on [A] , the 
rising article, we must spend a so many times larger or smaller 
sum on [B], the fallmg article. Or if we represent the total 



MATHKMATICAI. RELATIONS IGNORED 271 

sum of money to hv spent by S, and the sum to he spent on 
[A] l>y .s'a, we liave 






whence 

S<(1-^/). 

and representing' the sum to be spent on [B] 1)V s^^, we easily 
obtain 

For example, suppose A rises in price from l.UU to l.iJi* and 
B falls from 1.00 to .UU. Here all the employers of the above 
arguments would at once conclude that the level of prices has 
risen and money depreciated — the harmonic averagist amongst 
them. Yet it is possible with the same sums of money to get 
the same total (juantity of the goods at both periods. For here 

.fM)(1.99-l) .9801 

1.99(1 -.99) .0199 ^•-^^-'• 

Thus if we spend 100 money-units on [A] at each period we 
get at the first period 100 A and at the second 50.2512 A, and 
if we spend 4925.1 2| money-units on [B] we get at the first 
l)eriod this quantity of B and at the second 4974.8737 ; and 
the total quantity of [A] and [B] bought at the first period is 
5025.121, and the total quantity of [A] and [B] bought at 
the second is the same. 

It is plain that if at both periods we devote all our monev to 
purchasing the article which has risen in price, we get the ex- 
treme diminution in the total quantity purchasable at the second 
period compared with the total quantity purchasable at the first ; 
and if at both periods we devote all our money to purchasing the 
article which has fallen in })rice, we get the extreme augmenta- 
tion in the total quantity purchasable at the second })eriod com- 
pared with the total quantity purchasable at the first. Between 
these extremes it is evident, by the law of continuity, that there 



272 ARGUMENTS FOR THE OTHER AVERAGES 

must be some distribution of our speudings which will give 
neither diminution nor augmentation in the total quantities. 
This is the distribution indicated by our formula. It is evident, 
further, that if we spend more on the article rising in price than 
the proportion indicated by the formula, we get a smaller total 
quantity at the second period, and more and more smaller as we 
depart from this proportion, up to the limit when we spend all 
on this article. And reversely if we spend more on the article 
falling in price than the proportion indicated, we get a larger total 
quantity at the second period, increasing up to the limit when 
we spend all on this article. Thus between the two limits there 
is an infinity of total quantities that may be purchased with 
exactly the same sums of money at both periods. 

§ 5. In the second case, let the mass-unit used for each class 
be the same as m the preceding case, that is, the mass whose 
price at the first period is one money-unit, and again let the 
prices of these mass-units of [A] and [B] at the second period 
be a^ and /3./ respectively ; but let the number of the mass- 
units of [A] to be purchased at both periods be represented by 
x' , and the number of the mass-units of [B] to be purchased at 
both periods be represented by y' . Then the sums of money 
needed to purchase x' A is at the first period x' money-units 
and at the second x' a,^ money-units ; and the sum of money 
needed to purchase y' B is at the first period y' money-units and 
at the second y' ^^ money-units. Assuming that the total sums 
of these smns are the same at both periods, we have 

x' ^y' = x'a^ + ^';9/, 
which gives us 

^ ~ 1 - /v ■ 

And now again, if we take x' as a quantity-unit, we have 



which means that for every quantity of [A] we purchase we 
must purchase a so many times larger or smaller quantity of 



MATHEISIATICAL HET.ATIOJSS TC4NORED 273 

[B] (or must spend so inuch more or less money on [B] tlum 
on [A] at the first period). Or if we represent the total quan- 
tity of both classes by (^, and the quantity to be purchased (»f 
[A] bv q ., we have 

whence 



'L = 



N 



and re])resenting the quantity to be purchased of [B] by q^, we 
easily obtain 

Q« - 1) 

As these expressions give the quantities of equivalents of the 
first period, they also represent the sums that must be spent at 
the first period. 

Thus in the above numerical example, in which A is sup- 
posed to rise in price from 1.00 to 1.99 and B to fall from 1.00 
to .99, and in which the arithmetic averagist would probably 
see only a rise in the general level of prices, it is possible to 
get the same quantities of [A] and [B] at both periods. For 

here 

,, 1.99-1 .99 ^^ 

'^ ~ 1 - .99 ~ .01 ~ ' 

that is, if we purchase 100 A at each of the periods, we s})end 
for it 100 money units at the first period and 199 at the sec- 
ond ; and if we purchase 9900 B at each of the periods, we 
spend for it 99.00 money-units at the first period and 9801 at 
the second ; and so at the first period we spend for the quanti- 
ties of [A] and [B] together the total sum of 100 + 9900 
= 10,000, and at the second the total sum of 199 -f- 9801 
= 10,000, or in other Avords, for 10,000 money-units we can at 
each period purchase 100 A and 9900 B. 

Here, too, it is plam that if at both periods we purchase only 
the article which rises in price, we have the extreme augmenta- 
tion in the tcttal sum of money needed to purchase the same 
18 



274 ARGUMENTS FOR THE OTHER AVERAGES 

quantity at the second period compared with the total sum 
needed to purchase it at the first ; and if at both periods we 
purchase only the article which falls in price, we have the ex- 
treme diminution in the total sum of money needed to purchase 
the same quantity at the second period compared with the total 
sum needed to purchase it at the first. Between these extremes, 
again, it is evident, by the law of continuity, that there must be 
some distribution of our purchases which will require neither 
augmentation nor diminution in the total sum of money needed 
at the two periods to purchase the same quantities. This is the 
distribution indicated by our formula. Again it is evident that 
the further we depart from this distribution of our purchases by 
purchasing at both periods more of the article rising in price, 
the greater will be the total sum of money required at the sec- 
ond period compared with the first, up to the limit when we 
purchase only this article ; and reversely, by purchasing at both 
periods more of the article fallmg in price the smaller will be 
the total sum needed at the second period compared with the 
first, up to the limit when we purchase only this article. Thus 
between the two limits there is an infinity of total sums that 
may be needed to purchase exactly the same quantities of articles 
at both periods. 

Evidently uo arguments have validity that rest merely on 
some possibilities — either on the potentiality of certain sums of 
money to purchase quantities of goods, or on the potentiality of 
certain quantities of goods to command sums of money. 

III. 

§ 1 . Of the arguments as hitherto employed the defectiveness 
may be cured. It consists primarily in the neglect of weight- 
ing. Therefore we must first of all introduce into them con- 
sideration of weighting. 

Disregard of weighting we have seen to be the fault with the 
objections that have been urged against the employment of any 
average as an mdicator of constancy or variation of exchange- 
value. Some persons have played with the same price varia- 
tions, differently representing them, and really representing them 



COIJIIECTION OF THKHl DEFECTS 275 

SO as t(» express difterent weightings; and getting different re- 
sults, have denounced the wliole subject as intractable. Siniihir 
disregard of weighting has been the besetting sin in all the argu- 
ments for each of the averages, and the (^ause why none has been 
convincing. 

Tluis in the (U)ntroversy a])ove reviewed neither Jevons nor 
Laspe}'res sought for any more data than the mere variations of 
prices. Their dispute was very much as if they posited that 
some men are six feet tall and others are five feet tall, and 
quarreled over the average tallness. Factors necessary for the 
solution of the problem were absent ; yet they blissfully went on 
with the attenn)t to solve it. They each relied on various possi- 
bilities in the purchases, instead of requiring to have given, as 
real or as suppositional data, what were the actual purchases. 
Or if they did think they were agreeing upon the use of even 
weighting, they each conceived of this differently. Laspeyres 
supposed equal mass-quantities to be purchased of both articles, 
and constantly so at both periods (and also equal thaler's worths, 
but only at the first period). Jevons, when he suggested the 
harmonic average, supposed equal suras of money to be spent 
on both classes, and constantly so at both pei'iods ; but when he 
advocated the j^ -^ometric average, he gave no hint Avhat concei)- 
tion he had of even weighting. Such argumentation disparages 
tlie whole subject of exchange- value mensuration, and strengthens 
the hands of its opponents. 

In general, writers on the subject have made arguments for 
the different averages without regard to weighting, and they have 
made arguments — if arguments they deserve to be called — for 
several kinds of weighting without regard to the averages. They 
have never combined them. Up to this point separate argu- 
mentation has been employed in this work also. We now need 
to make the combination. 

§ 2. It is plain that in the argument made by the harmonic 
averagist the weighting is according to the constant sums of 
money that are supposed to be devoted at both periods to purchas- 
ing variable quantities in the different classes. 

This being so, certain peculiarities arise, due to the other 



276 ARGUMENTS FOR THE OTHER AVERAGES 

factor in this case, namely the sizes of the mass-units used, whose 
numbers are the mass-quantities between which compensation is 
desired by their equality. These peculiarities may here be 
briefly indicated. 

Let x^ and x.^, y^ and y.^, represent the numbers of mass- 
units, whatever these be, of the classes [A], [B], , purchased 

at the first and at the second periods respectively with the con- 
stant sums a, b, The only conception we as yet have of 

the mass-quantities are represented by these symbols x^, x^, y^, y^, 
And so the argument from compensation by equal mass- 
quantities for constancy in the exchange-value of money, or for 
variation in this exchange-value by the variation in the total 
mass-quantities, calls for formulation as follows, 



Mo2 ^ ^2 + 3/2 + 

Moi x^ + 2/1 + 
whence, by inversion, 

Pj _ ^x + 3/1 + •••• 

Pi '^2 + 2/2+-- 



(1) 



(2) 



But unless we specified that we had already selected the proper 
mass-units for this purpose, that is, if we only used the ordinary 
mass-units such as may be variously used by merchants, it is 
evident that in such formulae the result would be variable ac- 
cording to the sizes of the mass-units we happened to use,^ and 
we should be committing the same sort of absurdity as committed, 
from the other side, by Dutot.^ We must add a restriction to 
these formulae by including in them the method of selectmg the 

mass-units. Let a^ and a^, /9^ and ^^, be the prices at the first 

and at the second periods respectively of the mass-units of [A] , 

a 

of [B], which we do happen to use. Then x^= —and 

a b b 

3^2 = — , 2/1 = y and 2/2 = y , and so on. If now we converted 



« 



^r ''~^.' 



the last formula into this, 

1 See Chapt. V. Sect. III. ?5. 

2 See Chapt. V. Sect. VI. § 3. 



CORRECTION OF THEIR DEFECTS 277 



a b 



Pj ~a b 



(3) 



we should still have no single determinate result.'^ The prin- 
ciple of selecting the mass-units is still missing. A simple sug- 
gestion is that we should select mass-units that are equivalent. 
But then the questions arise, Equivalent at one of the periods 
only ? and at which ? or over both the periods together ? Now 
without consideration, without offering a reason, without argu- 
ment, people have agreed upon the convenient j)ractice of em- 
ploying mass-units that are equivalent at the first period. This 
being done, and in the still more convenient form of employing 
mass-units that are equivalent to the money-unit at that period, 
so that a^ = ^9^' =: = 1, the last formula reduces to 

P, a + b -I- 



P^ ~ a b 

or by using n" to represent a -f- b -|- •••• 



P., 

P, ~ 1 / 1 .1 



and, by restoring the price variations to their original forms, 



(4) 



^2 1 



l(a':^ + bf+ ) 



These two are formulae (15, i) and (16, i) given in Chapter V. 
Section V. § 1, as the formulae for the luvrmoniG average of 
price variations with weighting according to a, b, 

This is why this argument, applied in this way, is an argu- 
ment for the harmonic average of price variations. But there 
is no necessity for it to be applied in this way, for which no 

3 Cf. Chapt. V. Sect. VI. § 2. 



278 ARGUMENTS FOR THE OTHER AVERAGES 

reason has been oifered, so as to become an argument specially 
for the harmonic average. Applied to other mass-units, it may 
become an argument for other averages. 

Thus suppose we select mass-units that are equivalent to the 

money-unit at the second period. Then «/ = ^9/ = = 1, 

and the above formula (3) reduces instead to 

a b 

p^ _ «7 "^ g "^ 

P, - a-Hb+ ' 

or, by using n" to represent a -)- b 4- , and restoring the 

price variations to their simpler forms, 

h^{^.^'h > ^'y 

which is formula (16, 2) given in Chapter V. Section V. § 1, as 
the formula for the arithmetic average of price variations with 

weighting according to a, b, 

Or again we might equally well, for all we have as yet heard 
to the contrary, and perhaps better, select mass-units that are 
equivalent over both the periods compared. The method of 
getting these has been examined in Chapter IV. They are 
such that the geometric means of their prices are equal. This 
suggests some relationship with the geometric mean or average 
of price variations. Now we do not find any exact connection 
here with the geometric average — that is, when we are dealing 
with several classes or with uneven weighting. But when we 
deal with only two classes evenly weighted (that is, in this case, 
equally large at each period), so as to be able to employ the 
geometric mean of the price variations, the connection is per- 
fect. For now we have a = b, wherefore we may represent 
them each as 1 ; and from the nature of the mass-units (because 
of whose peculiarity we may distinguish their prices and mass- 
quantities by doubly priming them) Ave have a^'o.^' = i^^'^^'f 

8 "3 " B "d " 

whence a/' = ^^jj- and a/' = --^ • By substituting the 

first of these values in formula (3) we get 



CORRECTION OF THKIR DHriXTS 279 






(<^>) 



and l)v substituting the so(!ond, 



5," + '^. 



i'\ 



Hence ^ve liave, together, 



whence 



(') 



P^ 


= 


«./' 

p 




p^ 


= 


«./' 


+ /V 


p. 


«/' 


+ /5/' 



(8) 



which we should also have obtained directly, had we simul- 
taneously substituted the two values/ (Here in parenthesis we 
may notice that, as .r/'«/' = .r.^'a," = a = 1 and y('^(' = Vi'^^i' 

= b = 1, we have «/' = /^,, <' = V,, A" = J,,, mid ^- = 

•' 1 •*2 "1 

y, ; wherefore by substituting these values in the formulte ((Jj, 
(7), and (8) we get 

P; :.," ^," -^' + 2//'' ^'^ 

the last part of which we knew already, according to the hy- 
pothesis.) Now from the above combination of formulae ((3) and 
(7) we also form 

/p,y^ a- ,V' 

VpJ iV'*<'' 

whence we draw 



P \(j " B " 



p 

* This is Dutot's method, which, therefore, is rational in this one special ease. 



280 ARGUMENTS FOR THE OTHER AVERAGES 

^vhicli is the formula for the geometric mean of the price varia- 
tions of the two classes, with even weighting, being like the 
formula (14, s) in Chapter V. Section IV. § 2. 

■Thus, when attention is paid to the weighting, this argument 
from compensation by equal mass-quantities turns out to be still 
an argument equally ^vell, as yet, applicable either to the har- 
monic or to the arithmetic means or averages or to the geometric 
mean. Therefore in order to decide what average or mean it 
really favors, — or if it favors the geometric mean in the case of 
two equally important classes, what is the method it favors in 
the case of many variously important classes, — we must pay at- 
tention not only to the weighting, but also to the mass-units it 
is proper to use. We must search for a reason ^\^hy one set of 
mass-units is to be preferred, conductmg an investigation which 
has hitherto been neglected. 

§ 3. In the argument made by the arithmetic averagist the 
weights cannot be according to the constant mass-quantities that 
are purchased at each period by variable sums of money (unless 
we know some way of properly selecting the mass-units for 
this purpose), nor can they be merely accordmg to the sums of 
money spent on them, because these are different at the two 
periods. Yet they must be somehow connected with these sums 
of money, as we have already examined in Chapter IV. The 
formula which represents the conditions to which this arguments 
is applied and expresses its treatment of them is easily seen to 

be either this, 

P n. _u h _L 

(11) 



P, ^, + \ + 



P^ B., + \ + 

or, more definitely, 

Pi xa^ + yi3, + ' ^ "^ 

in which x, y, represent the numbers of times the ordinary 

commercial mass-units of [A], [B], , Avhose prices are 

a^ and a^, ^^ and ^.^, , at the first and second periods re- 
spectively. Now this last is the formula for Scrope's method, 
discussed in Chapter V. Section VI. § 4, and there shown to 
represent, or to be identical with, the arithmetic mean or average 



COIMtKCriON OF THKFR DEFECTS 281 

of tlio price variations with wciohtiiio- according to the sums of 
money spent on the constant mass-quantities in the different 
classes at the^r.s-^ period. I^ut that analysis was not complete. 
We shall later find that tliis formula equally well represents the 
hdniionic average or mean of the price variations with weighting 
according to the sums spent at the i^eeond period, and again, in 
some cases (namely when we are dealing with only two ecpially 
important classes), the geometric mean with (even) weighting ac- 
cording to the geometric mean of the sums at botli the periods. 

Thus, again in this case, with attention paid to the weighting, 
the argument is still applicable as well either to the arithmetic 
or to tlie harmonic means or averages or to the geometric mean. 
Here, however, we are no longer bothered by the (piestion of 
the mass-units to be used ; and yet we shall find that the selec- 
tion of the mass-units has had something to do with producing 
the appearance of this argument being more specially in favor 
of the arithmetic average. Here, too, we shall see that the ques- 
tion is not between different kinds of averages yielding different 
results, but between three different interpretations of, or three 
different ways of paralleling, one result yielded by one method. 

§ 4. Thus in general we have, not an argument for the har- 
monic average and an argument for the arithmetic average of 
price variations, as their employers have hitherto conceived them 
to be, but two arguments for either of the three averages, or 
means, applicable the one to one state of things and the other 
to another, — betAveen which arguments, therefore, when rightly 
confined each to its own field, there cannot even be contradic- 
tion, or antagonism. 

The fields to which these arguments are by their own natures 
confined remind us of the first two divisions in the question of 
weighting which we discussed in Chapter IV. Section V. But 
those two divisions in the question of weighting we recognized 
to be incomplete. Similarly these two arguments, correspond- 
ing to those two divisions, are incomplete. The argument made 
by the harmonic averagist supposes that Ave spend the same sums 
of money on every class at both periods in spite of the variations 
in their prices, which we rarely, if ever, do. The argument 



282 ARGUMENTS FOR THE OTHER AVERAGES 

made by the arithmetic averagist supposes that we buy the same 
quantities of every class at both periods in spite of the varia- 
tions in their prices, which we rarely, if ever, do. As a rough 
proposition, we — a community — generally spend more on articles 
that have risen in price and get less of them, and spend less on 
articles that have fallen in price and get more of them. At all 
events, we almost always spend oar money, and buy goods, in 
different proportions at any t"svo periods. Thus the more usual 
state of things is neglected by the arguments as made by the 
harmonic and the arithmetic averagists, except in a possible sup- 
plement to each argument by which the more complex state is 
reduced by curtailment to the one or to the other of the states 
required by these arguments. Then, as already remarked, the 
arguments become antagonistic ; but, so used, neither will have 
much claim for our respect. 

Now the third — the more usual, the more complex — state of 
things is really made up of each of the other two, in this way : — 
In the first state of things above described the particular mass- 
quantities are different at the different periods, and in the second 
state of things the particular sums of money are diiferent at the 
different periods ; and this double difference is precisely what ex- 
ists in the more complex state. 

Hence, even though the two states of things required in tlie 
two arguments are not likely ever to happen in reality, it is well 
for us to examine these arguments thoroughly, just as if their 
states were likely to occur ; because after reaching the right 
method for each of these states, we shall be in a position to com- 
bine them, and so form the right universal method, applicable 
to the complex states which generally exist. 

Thus our future work is mapped out for us. We must ex- 
amine the two arguments separately, each applied to its own 
state of things ; and then we shall seek what can be united of 
the two for the complex state. The three divisions of weight- 
ing, which we previously disposed of, so far as then possible, in 
one Section of one Chapter, now become divisions in the question 
of the averages, will occupy the next three Chapters. 



CHAPTER X. 

THE METHOD FOR CONSTANT SUMS OF MONEY. 



§ 1. Having examined in a general way the argnment from 
compensation by equal mass-quantities, we must now examine it 
in detail, applying it to many particular examples, for the pur- 
pose of discoyering in them a principle that will give us a clear 
indication of the true average or mean or method for measuriuir 
the constancy or variation in the exchange-value of money in 
the cases when constant sums of money are spent on every (^lass 
at both the periods com])ared.^ Also we may look for some 
crucial instances, serviceable as tests, that shall render our con- 
clusions demonstrative. We must survey first of all the differ- 
ent ways in which the argument is applicable to the different 
averages, or means, in order that we may later be in a position 
to judge in which application the argument is valid. 

By compensation by equal mass-quantities it cannot be meant 
to judge things by actual masses, counted in any of the usual 
weights or measures ; but the idea is to compare things by pro- 
portions of masses. If, for example, A, purchasable at the first 
period with one money-unit, be a quarter of barley, and B, like- 
wise then purchasable with a money-unit, be a bushel of wheat, 
a loss of one third on [A] at the second period is a loss of 2| 
bushels of barley, while a gain on [B] by a third is a gain of 

^ All that is said in this Chapter may be extended also to cases in which at 
both periods there are expended on all the classes sums in the same proportion, 
whether at the second period they be all smaller or all larger than at the first, 
providfd all the reasoning (and all the formula' later to be described, except two) 
be applied only to the sums of the one or of the other period — or, theoretically 
better, only to what is common to both periods. For the sums in excess, all i)eing 
in the same proportion, have no influence to alter the exchange-value of money 
already determined. 

283 



284 THE METHOD FOR CONSTANT- SUMS 

only ^ bushel of wheat. With two money-units evenly distrib- 
uted we could purchase at the first period 1 A -|- 1 B, or 9 
bushels of grain, and at the second f A -f |^ B, or 6^ bushels of 
grain — a loss m bulk. Or if A be one pound of copper and B 
ten pounds of iron, a loss of ^ on [A] is a loss of ^ pounds of 
copper, and a gain of ^ on [B] is a gain of 3J pounds of iron, 
so that from purchasing 11 pounds of metal our two money- 
units evenly distributed will come to purchase 14 pounds of 
metal — this time a gain in weight. Thus even the arithmetic 
equality m the compensation offered by the harmonic system 
disappears as a compensation by quantities of masses literally 
taken — or rather, taken at haphazard. The equality is system- 
atically obtained only by treating the masses of [A] and [B], 
whatever they be, purchasable with the same sum of money at 
some period or periods, as equal because they are equivalent. 
That is, these masses are conceived as equal, not as weights or 
capacities, but as exchange-values. For then the equal mass- 
quantities are conceived as composed of equal numbers of such 
equivalent individuals. A loss of one third on one article is 
supposed to be compensated by a gain of one third on another, 
even though this one third be but a small fraction by weight or 
bulk of the other one third, or even though it far exceed the 
other in weight or bulk, provided it be as much more or less 
valuable as it is physically smaller or larger, so as to be equiva- 
lent to the other, its greater preciousness making uj) for its lack 
of weight or bulk, or reversely, — at some period or periods. 
The argument is really in its proper form when it passes from 
the quantities purchasable to the powers of purchasing them, and 
claims that a loss of a third in the purchasing power of money 
over [A] (or its exchange- value in [A]) is to be compensated 
by a gain of a third in the purchasing power of money over [B] 
(or its exchange- value in [B] ). But as we measure particular 
exchange-values or purchasing poAvers of money by the quanti- 
ties of the things a constant sum will purchase, we may contmue 
to treat of the compensation by direct reference to the quantities. 
jSTow as we are dealing only with the proportions of the quan- 
tities, we have three ways of conceiving of the proportions, 



SCHEMATA 285 

when avo\vedly dealing with equally important classes : — either 
(1) as pfoj^ortions of variation //'o//; an equal condition at the 
first period, or ("2) as proportions of variation to an equal condi- 
tion at the second period, or (3) as proportions of variation, in 
any two classes, in the one from a certain condition to another 
and in the other from the latter condition to the former. Tiie 
same three positions are obtained by supposing the arithmetic 
compensation by equal quantities to be in equal numbers 
of mass-units (ideally constructed for the purpose) that are 
equivalent (1) at the fird })eriod, or (2) at the .second })eriod, or 
(3) over both the two periods together. It is more usual to 
adopt the first of these methods of measurement and this is the 
reason why this argument from equal mass-cpiantities has seemed 
to be an argument specially favoring the harmonic average of 
price variations. But we must examine all three of these 
methods in turn. 

§ 2. Using mass-units that are cqiiicalcat at the fird jjeriod, 
we may construct the following schemata illustrative of the con- 
ditions when there is compensation by equal numbers of such 
mass-units. We may still at first confine our attention to two 
classes supposed to be ecpially large in some respect. In the 
schemata are supposed to be expended at each period, marked I 
and II, certain constant sums, which are stated on the right- 
hand side. On the left are stated the mass-quantities, that is, 
the numbers of these mass-units purchasable with these sums at 
their prices, also stated, at each period ; and on their right, or 
in the middle, are added the sums or totals of these mass-quan- 
tities. Thus, on mass-units equivalent at the first period we 
have compensation by equal mass-quantities when the price 
variations are to the simple harmonic extremes, as follows : 



I 100 A @ 1.00 100 B (5,', 1.00 — 200 
II 66fA@1.50 133iB@ .75 — 200 



100 for [A] 100 for [B], 
100 for [A] 100 for [B]; 



when the price variations are to the simple arithmetic extremes, 
as follows : 

I 100 A @ 1.00 33^ B (<i, 1.00 — 133* I 100 for [A] 33* for [B], 
II 661 A @ 1.50 66fB(W, .50 — 133* | 100 for [A] 33* for [B]; 



286 THE METHOD FOR CONSTANT SUMS 

when the price variations are to the simple geometric extremes, 
as follows : 

I 100 A @ 1.00 66t B @, 1.00 — 166| I 100 for [A] 66| for [B], 
II 66|A@l.oO 100 B@ .661 — 1661 I 100 for [A] 661 for [B].^ 

Here we have the peculiar features with which we are already 
familiar. In the first the numbers of mass-units are equal at 
the first period, and the compensation is by arithmetically equal 
variations in them — over equal distances away from this equal 
condition. In the second the numbers of mass-units are equal 
at the second period, and the compensation is by harmonically 
equal variations in them — over equal distances going toward 
this equal condition. In the third the numbers of the mass- 
units alternate and change places, traversing not only equal dis- 
tances, but, so to speak, the same road, so that in them the com- 
pensation is by geometrically equal variations. 

The universality of these relations existing in the first of 
these particular examples has been demonstrated near the end 
of the preceding Chapter. They all admit of demonstration by 
means of one of the formulse discovered earlier in the same 
Chapter — in § 4 of Section 11. In the formulation there made 
the mass-units were supposed to be equivalent at the first period, 
so that it is applicable here. Let the price of A always be sup- 
posed to rise from 1.00 to «./. Then if the price of B falls 

a ' 
from 1 .00 to the liarmonic extreme, it will fall to ^^ — r^- — zr : if 

2a/ - 1 

to the arithmetic extreme, it will fall to 2 — «./ ; if to the geo- 
metric extreme, it will fall to , • Supplying these values of ^^ 

2 

in the formula 

- Here in all the schemata the differences in the numbers of A and of B are the 
same, and the totals are different. Arrangement can be made so that the totals 
would be the same, as follows for the second and third schemata : 

I 150 A @ 1.00 50 B @ 1.00 -200 I 150 for [A] 50 for [B], 
II 100 A @ 1.50 100 B @ .50 - 200 | 150 for [A] 50 for [B] ; 

I 120 A @ 1.00 80 B @ 1.00 -200 I 120 for [A] 80 for [B], 
II 80 A @ 1.50 120 B @ .66S - 200 | 120 for [A] 80 for [B] . 

But here the differences are different. The arrangement employed in the text is 
more perspicuous. 



SCHEMATA 287 

~<(i-/V)' 

MT find, wlicn the price variations are to the liarmonie extremes, 

b' = l; 

when they are to the arithmetic extremes, 

2 - «./ 
b' = ^- • 

when the\' are to tiie geometric extremes, 

The first of these ex])ressions means tliat when the price varia- 
tions of two classes are to tlie opjM)site harmonic extremes, in 
order to have compensation by etjiial numbers of mass-units 
e([uivalent at the first period we must spend our money evenly 
on the t^^•o classes at both periods ; wherefore we must purchase 
at the first period equal numbers of their mass-units, and at the 

1 2«./— 1 . 

second i)eriod , A and — , B, which are arithmetic extremes 
«., (I-.' 

around 1, since half their sura is 1. . The second means that 
when the price variations are to the opposite arithmetic ex- 
tremes, in order to have compensation by e(}ual numbers of such 
mass-units we must spend at both periods for every 1 M on [A] 

2 a' 

2 jyj- ^jj rj^-| ^vherefore the numbers of the mass-units 

purchased at the first period are in these proportions, and at the 

2 - «/ 

1 ~^r^ 1 

second i)eriod they are , A and -; — = — ; = , B, that is, they 
(J-l 2 — a J a J ' -^ 

are equal numbers at this period ; and now 1 , = — , — 

2 — «./ . . . , 

— , " , which shows that the numbers at the first period are 

arithmetic extremes around the conunon number at the second. 
The third means that when the price variations are to the op- 



288 THE METHOD FOE CONSTANT SUMS 

posite geometric extremes, iu order to liave compensation by 
equal numbers of such mass-units, we must spend at both periods, 

for every 1 ]\I on [A] — ~, ]M on [B] ; wherefore the numbers of 

the mass-units purchased at the first period are iu these propor- 

1 

1 ~^' 

tions, and at the second they are — -. A and -7- = 1 B, so that 

these numbers alternate over the two periods. ^ We may there- 
fore use the above particular examples, and the relations found 
in them, as universally illustrative. 

Now if on the price variations iu them supposed we employ 
the harmonic average on the first, the arithmetic on the second, 
and the geometric on the third, all with even weighting, we get 
in every instance an indication of constancy. But we have no 
right to use even weighting in every case. The only reason 
we can find for using even weighting in all these cases is that in 
all of them the same numbers of these mass-units are purchased 
at some period or periods, — iu the first at the first period, in the 
second at the second, in the third at each of the periods alter- 
nately. But of course there is nothing to recommend such a 
combination of weighting and of averaging, it being remembered 
that these mass-units are equivalent only at the first period.* 

^ In this last case if tlie total sum to be spent on the two classes is 2.00, we find 

that at both the periods we must spend ", ^ , M on FAl and — r^-r M on [B], 

getting these numbers of A and B at the first period, and at the second the reverse. 
These figures, rather curiously, are the harmonic means, the first between 1 and 

a,', the second between 1 and — ; i^ = ^.^)_ They are also arithmetic extremes 

around 1. (Thus the harmonic means between unity and geometric extremes 
around unity are arithmetic extremes around unity. ) 

* Another identity in the results deserves notice. In all three cases if we use 
the harmonic average with weighting in each case according to the numbers of 
these mass-units at the first period, or if we use the arithmetic average with 
weighting in each case according to the numbers of these mass-units at the second 
period, or if we use the geometric mean with weighting according to the geometric 
means of the numbers of these mass-units at both periods (whenever it happens 
that these are equal), all these means (and tlie first two averages in all cases) 
always give, applied to the same cases, identically the same results, (and in the 
more complex cases the geometric average, with its weighting, generally gives 



SCHEMATA 289 

Tlic only system of weighting recjuircd by this ;ir<;unK'iit from 
compensation by ('(jiuil (iiiantitics is the system of weighting ac- 
(iording to the constant snms devoted to purchasing each class 
at both periods. There is no use claiming that because the 
mass-quantities would then be different, we ought to correct this 
difference in the two worlds compared by reducing the mass- 
(piautities either by taking only the smaller (piantity in any class 
at either period (so as to get the largest (juantity common to 
l)oth the periods) or by taking some average of them. For 
then the sums })aid for such reduced or averaged constant quan- 
tities would be different, and the two worlds would be no more 
alike tlian before. The truth is, we are now engaged in meas- 
uring the variation in the purchasing power (exchange-value) 
of given sums of money by the variations — not of prices (ex- 
cept as these indicate the others) — but of the mass-quantities 
purchased. Hence the weighting is not to be according to 
the mass-quantities, but according to the sums devoted to pur- 
chasing them. The economic worlds we are considering are 
really made up of these sums, Avhich are supposed to be the 
same at both periods ; and what we are measuring is the varia- 
tion in their purchasing powers. In the next Chapter, when 
we have under examination the argument from compensation by 
equal sums, the economic worlds will be made uj) of the mass- 
quantities (but still to be conceived as exchange-values), the 
variables then being the sums they will (command, indicative of 
their varying exchange-values in money and of money's inversely 
varying exchange-values in them. 

Here, then, the weighting being according to the constant sum.s 
expended on eacli class at both periods, it is only the first of the 
above schemata in which even weighting can be used. In the 

/ 2 — a '\ 
second the weighting is 1 for [A] and J [ = /^ \ for [B] ; 

very nearly the same results as the other two with theirs). Here in the first ivn* 
averagings the periods of the weightings are inverted from those given as general 
principles in Chapt. VIII. Sect. I. § 7. This is because tlie variations of the 
mass-quantities are the inverse of the price variations. A similar identity will 
oecijpy our attention in the ne.xt Chapter (where the periods of the weightings 
are the proper ones, and tlie weighting itself is proper). At present we Imve Little 
interest in these relations. 

19 



290 THE METHOD FOR CONST A^S'T SUMS 

and ill the third it is 1 for [A] and ^ (= A for [B] . Now 

if we use the harmonic average on each of these cases with 
these weightings, we again always get an indication of constancy. 
But if we use the other averages with these weightings, we get 
very diiferent results. The arithmetic average, in each case 
with its proper weighting, indicates for the first a rise of 12|^ 
per cent., for the second a rise of 25 per cent., and for the third 
a rise of 16f per cent. And the geometric average, in each 
case with its proper weighting, mdicates for the first a rise of 
6.066 per cent., for the second a rise of 14 per cent., and for 
the third a rise of 8.44 per cent. 

As neither of these other two systems pretends to claim con- 
stancy when the proper weighting is used with each average, it 
is not easy to compare the different averages on these schemata. 
To compare them we shall want rather the schemata in which 
equal constant sums are spent on every class, so that even 
weighting may be used in each case, wherefore each average, the 
price variations being as before, will indicate constancy. 

§ 3. Let us noAV notice the same price variations with equal 
numbers of mass-units gained and lost when these are equiva- 
lent at the .second period. It is easy to adapt the preceding 
schemata to these new mass-units. We need to change only the 
mass-unit of [B], which we shall represent by B'. It must be 
remarked that we are not rearranging the presentation of the 
same facts as in the previous schemata, but we are presenting 
different facts. The schema for the harmonic price variations is 



100 for [A] 200 for [B], 
100 for [A] 200 for [B]; 



I 100 A (a:, 1.00 100 B^ fol 2.00 — 200 
II 66i A Gi 1.50 133A B^ 01 1.50 — 200 

that for the arithmetic price variations, 

I 100 A @ 1.00 33i B' %, 3.00 — 133^ 
II 665- A @ 1.50 661 B^ @, 1.50 — 133^ 

that for the geometric price variations, 

I 100 A @ 1.00 66* B^ qi, 2.25 — 166f I 100 for [A] 150 for [B], 
II 66;i A r^/ 1.50 100 B' (^^ 1.50 — 166S I lOOfor [A] 150 for [B]. 



100 for [A] 100 for [B], 
100 for [A] 100 for [B] ; 



SCJIEMATA 291 

\lvvv the xai'vliiii' niiinbci's of the iiiass-units arc the same as 
ill tlu' previous schemata,' hut their ])ri('cs being variously 
hitiher iu caeii case, tlie constant sums expended on [B] are in 
each case variously different. That the above noticed peculiar- 
ities in regard to the numbers of the mass-units iu this rear- 
rangement are universal, could easily be proved in a manner 
similar to that before used, by finding first the general formula 
for the cases when, ji., ecpialling a.„ fi^ is the figiu'c that has 
varied harmonically, arithmetically or geometrically,'' and ajiply- 
ing this as before. 

Here again the harmonic average ap})lied to the first (^ase, the 
arithmetic applied to the second, the geometric ap])li('(l to the 
third, each with even weighting, all indicate constancy of the 
general exchange-value of money, l^ut in this schematization 
it is only the second case, in which the price variations are 
arithmetic, that has a right to the use of even weighting. In 
the first case the weighting is 1 for [A] and 2 for [B] ; and 
with this weighting the harmonic average indicates a fall of 
|)rices by 10 per cent. And in the third the weighting is 1 for 
[A] and 1| for [B]; and with this weighting the geometric 
average indicates a fall of prices by 7.79 per cent. But, always 
used with the proper weighting, the arithmetic average indicates 
constancy in every case. Therefore also these schemata are not 
suitable for comparing the averages ; and when we reada])t them 
all to the same (even) weighting only the second will remain. 

§ 4. Lastly we wish to schematize the compensation by ecpial 
mass-quantities when, in the same price variations, the mass- 
units are equivalent over both the periods together. As A is suj)- 
posed in all the cases to rise from 1.00 to 1.50, its (geometric) 
mean price is always 1.2247. Therefore the (geometric) mean 
price of B" is desired always to be 1.2247. This is obtained 
in the following schemata — for the harmonic j)ricc variations : 

5 Hence what is stated in the last note is still applicable also to the cases with 
these mass-units. And the numbers of these mass-units are no better criteria of 
weighting than the numbers of the mass-units thei-e used. 

^ This formula, with a^' at 1.00 and with a taken as the unit sum, is 






in which /3j = a,'. 



292 



THE METHOD FOR CONSTANT SUMS 



I 100 A @ 1.00 
II 66f A @ 1.50 



100 B^''@ 1.4142 — 200 
133JB^''@ 1.0606 — 200 



100 for [A] 
100 for [A] 



141.42 for [B], 
141.42 for [B], 



(in which 1.4142 : 1.0606 :: 1.00 ; 0.75, and 1.4142 x 1.0606 
= 1.00 X 1.50); for the arithmetic price variations : 



1100 A @ 1.00 SS^B'-'i 
II 66fA@1.50 66fB'^ 



1.732 — 133J 

.866 — 133 J 



100 for [A] 
100 for [A] 



57.73 for [B], 
57.73 for [B], 



(in which 1.732 : 0.866 :: 1.00 : 0.50, and 1.732 x 0.866 = 
1.00 X 1.50) ; for the geometric price variations : 



I 100 A @ 1.00 
II 66f A @ 1.50 



66|-B'^@ 1.50 — 1661 
100 B'^@ 1.00 — 166f 



100 for [A] 
100 for [A] 



100 for [B], 
100 for [B]. 



Here also the numbers of the mass-units are the same in these 
still other circumstances as in the preceding two sets of sche- 
mata ; "^ but with still other constant sums expended on [A] and 
[B] . By the same method of proof the above-noticed peculiar 
relations between these numbers can be proved to be universal.® 
Of course, as before, with even weighting, the harmonic average 
of the price variations in the first case, the arithmetic average 
of them in the second, and the geometric mean of them in the 
third, all indicate constancy. But here it is only in the third 
that even weighting is proper. In the others, each average 
with its proper weighting, gives a various result — the harmonic 
in the first indicating a fall of prices by 5.41 per cent., and in 
the second the arithmetic indicating a rise of prices by 13.33 
per cent. Therefore again we must re-adapt the schemata, only 
the third here given being serviceable. It may be added that 
in the other two cases, with their proper weighting, the geo- 
metric average yields results indicating slight divergences from 
constancy (in the first a fall of prices by 0.06 per cent., in the 
second a rise by 0.3 per cent.). The reason for these diver- 
gences is already known. Their meanmg will be explained later. 

^ Hence again the statement in Note 4 is applicable also to the cases with thtse 
mass-units. But now the numbers of these mass-units are good criteria of weight- 
ing, as we shall see presently. 

* The general formula for the relation between the sums, still with a^ at 100 
and with a as the unit sum, is 

ao' -1 
b' 



/3l-/^2 



in which /Si/'Jo = ao 



TiiK (ii-;()>ri:TRic mean proved 



293 



II. 

^ 1. When the ar*>iinu'iit from compensation by equal mass- 
quantities is made an argument especially for the harmonic 
average of price variations, and the harmonic price variations, 
that are su})i)osed to represent constimcy are contrasted with the 
other variations in a manner detrimental to the argument for 
tiiem, we have, for the simplest cases, which use even weighting 
throughout, the following schemata — for the harmonic; price 
variations : 



1 100 A Ov, 1.00 100 B («\ 1.00—200 
II mri X (ay \.-,{) Vm,¥,('v, .75—200 

foi- the arithmetic j)rice variations : 

1 lOo A 0^1.00 100 BO/, 1.00— 200 
II 60^ A('>, 1.50 200 \\(«': .50—2661 

for the geometric price variations : 



100 for [A] 100 for [B], 
100 for [A] 100 for [B]; 



I 100 A@, 1.00 
II 66s A (^,1.50 



100 
150 



B(,,, 1.00 —200 
B(^^ .661— 216f 



100 for [A] 
100 for [A] 



100 for [A] 
100 for [A] 



100 for [B], 
100 for [B]; 



100 for [B], 
100 for [B]. 



Here, the mass-imits all being equivalent at the first period, 
there is compensation by ecpial mass-quantities (ecjual numbers 
of these mass-units) only in the case of the harmonic price vari- 
ations, so that it is only in this case that the purchasing power 
or exchange- value of money seems to be constant, wherefore the 
harmonic average seems to be the right one, as it alone indicates 
constancy in this case ; while in the other price variations the 
compensation by mass-quantities is, in the arithmetic price vari- 
ations, for every one third lost on [A] a gain of one whole on 
[B], which seems very much too much, and in the geometric 
price variations for every loss of one third on [A] a gain of 
one iialf on [B], which still seems too much, wherefore the 
arithmetic and the geometric averages or means seem to be 
wrong because each in its own case, with the proper Aveighting, 
indicates constancy. It is wholly and solely on account of this 
special arrangement of the mass-quantities, due to the selection 
of the mass-units, which are equivalent at the first period, that 



294 THE METHOD FOK CONSTANT SUMS 

some persons have been led to suggest the harmonic average of 
price variations as the right one. 

But it is precisely because of this special arrangement that 
the harmonic average may be proved to be wrong. 

§ 2. The argument for the harmonic average claims that, 
the classes [A] and [B] being constantly equally large or im- 
portant over both the periods, if A and B, the equivalent mass- 
units at the first period, are equally precious at the first period 
(being equally heavy or bulky), a loss of one third on [A] is 
correctly compensated by a gain of the same mass (by weight or 
by bulk) on [B] ; or if they are not equally precious at the first 
period, a loss of one third on [A] is correctly compensated by 
a gain in the same proportion, namely by one third, on [B], 
this addition of ^ B being as much larger or smaller than J A 
as B was less or more precious than A at the first period. Evi- 
dently there is here a tacit assumption, which is belied by the 
very supposition itself, of continuance at the second period of 
the same relative preciousness as at the first period. A and B 
are supposed to be equivalent at the first period. Then J A and 
^ B are equivalent at the first period. Therefore at the first 
period if we distribute our purchases so as to get with two 
money-units f A and 1^ B instead of 1 A and 1 B, we have per- 
fect compensation in the gain of |^ B making up for the loss of 
^ A, because this gain is equivalent to this loss. But at the 
second period the supposition is that A, having risen in price to 
1.50, while B has sunk to .75, has become more valuable than 
B — in fact, just twice as valuable in this example. Therefore 
I B is no longer equivalent to ^ A, and the gain of ^ B is no 
longer sufficient to compensate for the loss of ^ A. 

Or let us put the question as one concerning purchasing 
power. If at the first period, possessing two money-units, Ave 
give up one third of our purchasing power (^ver [A] by using 
only f M to purchase [A] , and if at the same time, with- 
out any intervening changes of prices we gain one third in our 
purchasing power over [B] by using the ^ M saved from use 
on [A], there is perfect compensation ; for the counter-balancing 
purchasing powers are equal. But the supposition we are deal- 



TIIK (JEOMKTKIC MKAX I'ROVKD 295 

iuo; with is tluit at the second i)cri(Kl we can get only | A with 
1 M, and it is claimed that the compensation is good if we can 
get 1^ B with the other 1 M. Now nnder these eirenmstances 
it is true that the particular exchange-value of M in [A], or its 
|)arti('ular purchasing power over [A], has fallen by (»iie third, 
and that the particular exchange-value of M in [J5],or its par- 
ticular [)urchasiug power over [15] , has risen by one third, and so, 
if we were dealing with particular purchasing powers, this com- 
pensation might a})pear to be good. But we are really dealing 
with the general exchange-value of M, and so with its general 
purchasing ])ower. This has not fallen by one third because of 
M's purchasing only f A, unless M also purchases only | of 
everything else ; nor has it risen by one third because of M's 
purchasing 1^ B, unless M also purchases 1^ of everything else 
— which conditions are contradictory to each other and to the 
original sup])osition. We do not then as yet know ^^■hat the 
compensation ought to be. But we do perceive this, that when 
A alone rises in price and B alone falls in ])rice, M in gaining 
\ in (piantity of [B], which has fallen in exchange-value, has 
gained less in exchange-value than it has lost in losing ^ in 
(piantity of [A] , which has risen in exchange- value. The com- 
pensation oifered is no longer an addition of one third of an 
equal exchange- value or purchasing power in ])laee of a sub- 
traction of one third of a given purchasing power ; it is the 
addition of one third of a smaller-grown purchasing power in 
])lace of a subtraction of one third of a larger-grown pnrchas- 
ing power. 

In such cases, therefore, our money has fallen in })urchasing 
power or exchange- value — it has depreciated. And inversely 
the general level of prices has risen. The fall in the price of 
B to the harmonic term (a fall by J) is not great enough to 
comjjensate for the rise in the price of A (a rise by ^). And 
the harmonic average of the price variations is wrong in indicat- 
ing constancy. 

The fallacy in the argument for the harmonic average of prices 
from compensation by equal mass-quantities is that it employs 
a compensation which is good only on the assumption that the 



296 THE METHOD FOR CONSTANT SUMS 

relationship between the exchange-values or preciousness of the 
articles has remained the same at the second as at the first period, 
although the data argued upon preclude this continuance (ex- 
cept in case there are no irregular price variations). 

§ 3. On the other hand when the argument from compensa- 
tion by equal mass-quantities is applied to the arithmetic aver- 
age of price variations, there is a similar fallacy reversed. The 
arofument now tacitlv assumes that the conditions existing at the 
.second period existed also at the first ; for it employs the com- 
pensation which is good only at the second period. 

Thus in the above schema for the arithmetic price variations, 
where A and B are equivalent at the first period, the compensa- 
tion offered is a y-ain of one whole A for everv one third B lost, 
the gain being three times as large as the loss, measured in those 
mass-units. But we perceive that at the second period when 
the price of A has risen to 1.50 and the price of B fallen to .50, 
A has come to be, at the second period, three times as valuable 
as B, wherefore, at the second period itself, a compensation by 
a gain of three times as much of B as is lost of A is the proper 
compensation then. At this second period we are compensated 
by getting just as many times as much more of the fallen article 
than less of the risen as the former has become less valuable than 
the risen. ^ Here is a semblance of correctness in the position of 
the arithmetic averagist. The semblance is brought out more 
plainly by the following rearrangement of the schemata, in which 
merely the mass-units of [B] are altered, and consequently the 
numbers of them purchasable with the same sums of money, the 
facts represented being the same as before. The schema for the 
harmonic price variations is : 



100 for [A] 100 for [B], 
100 for [A] 100 for [B]; 



I 100 A @ 1.00 50 B^ (a) 2.00 — 150 
ir 66f A @ 1.50 66| B^ @ 1.50 — 133^ 

that for the arithmetic price variations : 

I 100 A @ 1.00 33i B^ (a) 3.00 — 133^ I 100 for [A] 100 for [B], 

II 66| A @ 1.50 66| B^ @ 1.50 — 133^ | 100 for [A] 100 for [B] ; 

1 The universality of tliis relationship, given conditions permitting of even 
weighting, is evident when we remember that the mass-quantities are in harmonic 
progression ; for in this progression around unity as the mean we know that 
l~a: b - I: : a: b. 



Tin-; (iKoMiyriMc mkan iMtovKD 297 

that f'oi" the iicoinctric |)ri('(' N'iiriiitions : 

I 1(>(» A Oo 1.00 44^ IV («j 2.25 — 144,^ I 100 for [A] 100 for [B], 
II (idj A Oi\ l..")0 mi \V Oi) 1.50 — 138^ ! 100 for [A] 100 for [B]; 

aiuono- which it is only the arithnu^tii^ price variations that gives 
coinpeusation hy arithmetically equal numbers of these mass- 
units ; while in the harmonic price variations the compensation 
seems to be very much too small (although it is really the same 
as before, and pro})er at the first period itself), and in the geo- 
nu'tric still too small ; wherefore now, by a mere change in the 
size of the mass-units, the arithmetic average of the price varia- 
tions, indicating constancy in this case, alone seems to be justi- 
fied in indicating constancy, and so seems to be the proper aver- 
age to use in all cases. 

The fault with this argument for the arithmetic average of 
price variations is that the compensation by mass-quantities 
which it offers is what ought to take })lace (it the second period 
alone. At this period when A = B' (or wdien A --3B), in an 
even spending of three money-units on [A] and [B] we get 
1 A and 1 B' (or '.\ B), and again in a spending of one third less 
on [A] and of one third more on [B] we gain just as much as 
M'c lose ; for we gain J B' (or 1 B) in place of ^ A lost. Here 
the compensation is perfect because the quantity lost is equiva- 
lent to the quantity gained in both the transactions compared 
(whether the masses gained and lost happen to be expressed in 
e([ual or unequal numbers of mass-units, according to the sizes 
of these). But in our suppositional case we are comparing a 
transaction at the first period when A (being equivalent to B) 
was equivalent only to |- B', with a transaction at the second 
period, at w^hicli alone A is equivalent to B' (or to 3 B). An 
t)ffered comiiensation of J B' (or 1 B) for a loss of J A at the first 
period would be three times too great. It is still too great, 
though not so much in excess, when it is offered at the second 
period in comparison with the first. We must reflect that J B' 
(or 1 B) has fallen in price, that is, it loas more valuable, and 
that A has risen, that is, it loas less valuable. The gain of a 
numerically equal amount (or three times as much) on an article 
which, although its mass-unit is equally (or three times less) vain- 



298 THE METHOD FOE CONSTANT SUMS 

able than the other at the second period, was three times more 
(or equally) valuable at the first period, is too great a gain. 
Similar would be the conclusion if we treated the subject from 
the point of view of purchasing power. Therefore, as this prof- 
fered compensation is too large for the quantity gained over the 
quantity lost, our money purchases too much to permit it to be 
stable : its purchasing power, and its exchange- value, has risen : 
it has appreciated. And inversely the arithmetic compensation 
by equal prices is too great for the loss over the gain, and the 
general level of prices has fallen, instead of being constant, as 
is wrongly indicated by this average. 

The generalization may therefore be made that in all cases 
when constant sums are spent on the classes at both periods, 
these sums being taken for the weights, the indication concern- 
ing the general level of prices offered by the harmonic average 
of the price variations is lower than it ought to be ; and the in- 
dication offered by the arithmetic average of the price variations 
is higher than it ought to be. 

§ 4. The true position must take account of the conditions at 
both the periods, and so the compensation must lie between those 
offered by the harmonic and the arithmetic terms, being larger 
than the former and smaller than the latter for the loss over the 
gain by mass-quantities. It is here that lies the compensaticm 
offered by the geometric mean and average. 

The schemata to illustrate the argument for the geometric 
mean are as follows — for the harmonic price variations : 

I 100 A Ov, 1.00 70.71 B'' @ 1.4142 — 170.71 I 100 for [A] lUO for [B], 
II 66| A @ 1.50 94.28 B^'' @ 1.0606 — 160.94 | 100 for [A] 100 for [B]; 

for the arithmetic price variations : 

I 100 A @ 1.00 57.73 W' @ 1.732 — 157.73 I 100 for [A] 100 for [B], 
II 66| A @ 1..50 115.47 W @, .866 — 182.13 | 100 for [A] 100 for [B]; 

for the geometric price variations : 

I 100 A @ 1.00 66| B'' @ 1.50 — 1661 I 100 for [A] 100 for [B], 
II 66| A @ 1.50 100 B'^ @ 1.00 — 1661 | 100 for [A] 100 for [B]. 

Here, the real facts represented being the same as in the two 
preceding sets of schemata, the only compensation by an equal 



TIIIO (iKO.MKTIlIC MKAN I'K(>\'K1) '29U 

luiniber of f/icsc niass-iiiiits is in tlic simple gcoinctric price 
variations, the compensation offered by the harmonic; price 
variations hein<>; too small, and that by the arithmetic too large. 
Thus we have the apj)earancc, e(iually good as such, that it is 
only the geometric mean of price variations that in indicating 
constancy in this case gives the right result. The question now 
arises. Are there any reasons why the ajipearance is better in 
this case than in the others? 

The reasons are plain. They flow from the })rineiples already 
examined in Chapter VIII., which principles are of direct a])- 
plication to our present subject. A^"e are dealing with a subject 
in which no matter how far the price of A rises, the quantity of 
[A] ])urchasablc with a given sum cannot fall to zero, and as B 
falls in price, the quantity of [B] can be limited by no figure 
short of infinity. Our subject then is suitable for the use of the 
geometric averaging of the price variations, in which constancy 
is shown when the quantities offered in compensation vary to 
the geometric extremes. Here the variations are all the reverse 
of each other. The mass-unit of [B] is no longer the equivalent 
of the mass-unit of [A] either at the first period or at the second 
period, l)ut it at the second period is equivalent to A at the first 
and it at the first is equivalent to A at the second. And in the 
variations of the numbers of the mass-units there is compensation 
not only by equal quantities but by equality of distance traversed 
over the same road in reversed directions. These inverted re- 
lations are universal (for all simple geometric price variations of 
two classes in opposite directions) when we have the conditions 
required by even weighting. It follows from them that there is 
alternation of .preciousness conjointly with, and oppositely to, 
the alternation of the mass-quantities. AVe lose one third on 
the whole of [A] while it gains in preciousness by half (reckon- 
ing from the first period), and (reckoning in the same way) we 
gain one half on the whole of [B] while it loses in preciousness 
by one third. This leads to the really fundamental reason, 
which is : — As the mass-unit here used of [B] is the equivalent 
of the mass-unit of [A] over both the periods, it is obviously cor- 
rect that we should gain as many of these mass-units of [B] as 



300 THE METHOD FOR CO^^STANT SUMS 

we lose of these mass-units of [A] . The compensation by mass- 
quantities is in this case perfect. 

The superiority of the geometric mean to the other means of 
price variations ma}' be shown more clearly by bringing together 
the three forms of the schemata on which the argument for each 
of the averages has relied. These are — for the harmonic average 
of price variations : 

1100 A @ 1.00 100 B(ai 1.00 — 200 
II 06|A(^rl.50 1331 B@, .75 — 200 



100 for [A] 100 for [B], 
100 for [A] 100 for [B]; 



for the arithmetic average of price variations 



1 100 A (a, 1.00 33i B^ @ 3.00 — 133i 
II 66f A @. 1.50 66f B^ g^ 1.50 — 133^ 



100 for [A] 100 for [B], 
100 for [A] 100 for [B]; 



for the geometric mean of price variations 



I 100 A @ 1.00 66| W (a) 1.50 — 166? 
II 66| A (a', 1.50 100 B'^ @, 1.00 — 1661 



100 for [A] 100 for [B], 
100 for [A] 100 for [B]. 



In the first of these it is plain that the mass-unit used of [B] 
is of smaller exchange-value than the mass unit of [A] over 
both the periods together. Hence in gaining only an equal 
number of these mass-units of [B] for the mass-units of [A] 
lost, we gain less exchange-value than we lose. Therefore our 
money has diminished in exchange-value or purchasing power, 
or has depreciated, and the general level of prices has risen ; 
wherefore the harmonic average of the price variations errs be- 
low the truth in indicating constancy. 

In the second it is plain that the mass-unit used of [B] is of 
greater exchange-value than the mass-unit of [A] over both 
the periods together. Hence in gaining an equal number of 
these mass-units of [B] for the mass-units of [A] lost, we gain 
more exchange-value than we lose. Therefore our money has 
augmented in exchange-value or purchasing power, or has ap- 
preciated, and the general level of prices has fallen ; wherefore 
the arithmetic average of the price variations errs above the 
truth in indicating constancy. 

But in the third the mass-unit used of [B] is of the same ex- 
change-value as the mass-unit of [A] over both the periods to- 
gether. Hence in gaining an equal number of these mass-units 



THE GEOMETRIC MEAN PROVED :}01 

of [B] for tlic mass-units of [A] lost, we gain exactly the same 
exchange-value as we lose." Therefore our money has remained 
constant in exchange-value or punrhasing {)()wer, and the gen- 
eral level of ])rices is also constant ; wherefore tlu; geonu'tric 
uKian is right in indicating constancy. 

The geometric mean also i"ightly indicates the variatioiis in 
the other cases. Let us go back to the first set of schemata in 
this paragraph, which, in the same order, represent the same 
states of things as the last. For there we have reduced the 
mass-unit of [B] to equivalence over both the periods with the 
mass-unit of [A] . In the first schema, where the price varia- 
tions are the harmonic, the purchasing power of money, or its 
exchange-value, relatively to these two classes, is evidently ac- 
cording to the totals of these equivalent mass-units which the 
given total sum of money will })urchase at each jjcriod, so that 

3Io., 1(50.94 . n. . p„ , 

we have ^jy-^ = _ ^y = 0.f)42S, mdicatmg a fall of 5.72 per 

cent., while the general price variation is the inverse, thus 

P, 170.71 . ,. . . ^ 

p = ,.,,, q J. "= 1.060b, mdicatmg a rise of 6.06 ])ercent. Aow 

the geometric mean of these price variations is — - = ^ x 

•> 
= = 1.0606, likewise indicating a general rise of prices by 

6.06 per cent. In the second, where the price variations are 

. , . , M,,. 182.13 . ,. . 

the arithmetic, we have v^" = -__ .,-r = 1.1.)4^, indicating a 

, P, 157. 7;i . . ,. . 

rise of 15.47 per cent., and p" = o.) i •> = O.X6()0, indicating a 

- These mass-units are not the economii" intlividuals deseribed, for the present 
supposition, in Chapter IV. Sect. V. ;J3, namely constant exchange-values with 
variable masses. But they are substitutes therefor ; and they are like the eco- 
nomic individuals described in Chapt. IV. Sect. V. jJ (i, which will Ije used in the 
supposition to be treated of in the next Chapter. They may very properly be 
used for the purpose they are here put to. To be sure, the weighting of the classes 
at each period cannot be measured by the numbers of tiiese mass-units they con- 
tain. Yet the weighting of the classes in the averaging of their price variations, 
over two periods together, is according to the geometric means of tlie numbers of 
such mass-units they contain at each period. Another example will be. given 
later (see below in Sect. IV. Note 1). 



302 THE METHOD FOE CONSTANT SUMS 

fall of 13.40 per cent.; and the geometric mean of the price 

"P I "^ 1 ^ 

variations is p^ = I o ^ >> ~ ~9~ =0.8660,likewise indicating 

a general fall of prices by 13.40 per cent. That the geometric 
mean of the price variations of two classes equally important 
over both the periods universally agrees with the inverse of the 

indication for -^" rendered by the numbers of the mass-units 

equivalent over both the periods purchasable at each period, has 
already been demonstrated near the end of the preceding Chap- 
ter in formula (10) there given. ^ Therefore the geometric mean 
of price variations, whenever it is applicable to cases in which 
for two classes of goods the same sums of money are spent at 
both periods, universally gives the right indication. 

§ 5. The fault with the argument from arithmetic com- 
pensation by equal mass-quantities, as it has generally been 
employed by the writers who have suggested or advocated or 
employed the harmonic average of price variations, has lain in 
the fact that utter neglect has been paid to the exchange- value 
of the mass-quantities whose loss and gain have been compared. 
To be sure, the mass-units have generally been chosen equiva- 
lent at the first period. But that has been due merely to conve- 
nience, and to the habit, itself due to convenience, of starting with 
units priced at 1.00 or at 100. The query why the compensa- 
tion should be by equal numbers of such mass-units, has never 
been raised ; wherefore the wrongness of such compensation has 
escaped notice. Possibly it has not been noticed that the so- 
called " quantities," or mass-quantities, as numerical figures, are 
not absolute figures, according to the masses given, but are de- 

^ P^urthermore, in both the above examples, in which the mass-quantity of [A] 
is 100 and the price of A 1.00, at the first period, the figure for the variation of the 
general exchange-value of money is identical with the hundredth part of the 
mass-quantity of [B] at the second period, and also it is the quotient of the price 
of B" at the first period divided by the price of A at the second ; while the figure 
for the general price variation is identical with the price of B" at the second 
period, and also it is the qiiotient of the price of A at the second period divided 
by the price of B" at the first. The universality of these relations, and of some 
others which it would be too long to notice, is also demonstrated by formulae (6), 

(7), and (fl) ( and by their inversions for vj- ) given near the end of the preced- 
ing Chapter. 



THE (iEO.METKlC MEAN I'liOVED ."503 

tcnniiu'd also hy the sizes of the inass-iinits in whicli the masses 
are iiR'asurcd. 

Many advocates not only of the harmonic, but even of the 
arithmetic average, have failed, as we have seen, at a still earlier 
stage. They would measure the general exchange-value of 
money, under the name of its "general purchasing power," l)y 
the mere mass-cjuantities that a given total sum of money can 
purchase, (tr has the power of purchasing, at each peri(»d, with- 
out regard cither to the mass-units used or to the acstual sj)end- 
iugsof our money at the two periods compared. Our own com- 
pleter analysis has shown that, when there is any change of 
prices, there is always, within fixed limits, a great variety of 
total mass-(|uaiitities that can be purchased with the same total 
sum of monty, according as diifcrent spendings of its parts (the 
same at both periods) be hit u])on. Therefore this argumenta- 
tion proves only a great variableness in "general purchasmg 
power " so conceived, whenever there is any change of prices. 
Thus if we look upon jnirchasing power as somethmg to be 
measured only by the (piantities of things that can l)e purchased? 
we see that for such a thing as the "general purchasing power" 
of money, so conceived, to remain stable, absolutely no change 
of prices must take place. So far Roscher and others are right 
in denying the possibility of stability of money in general pur- 
chasing power exce])t — to use Martello's })hrase — by " petrifac- 
tion " of the economic world. But they are wrong who ex- 
tend this denial to the general exchange- value of money. For 
it is evident that if later in any distribution of spendings one's 
money gets the same exchange-value which it got before, it gets 
the same exchange- value in any other distribution of spendings. 

It niight, however, l)e admitted that we can have such a con- 
cept as this of "general purchasing power" measured only by 
(quantities of things i)urchasable, if we desire to distinguish 
" purchasing power" from " exchange- value." For, as just re- 
marked, we know that it remains constant so long as all prices 
remain unchanged ; and we also perceive that if all prices rise 
or fall in exactly the same proportion, it rises or falls in that 
proportion — in the former case coinciding with exchange-value 



304 THE METHOD FOR CONSTANT .SU:MS 

in all other things and with exchange- value in all things, in the 
latter case with the former of these alone, in all other cases 
breaking up and disappearing. But such an evanescent concept 
is of no service in economics. 

The only proper way to conceive of general purchasing power 
is by the quantity of exchange-value it can purchase. It would 
be absurd to measure the lifting poAver of a derrick by another 
attribute, say volume, in the things which the derrick can lift, 
rather than by that attribute in them which is the essential ob- 
ject to which lifting power has reference, namely their weight. 
If the volume of all things were, and always remained, of equal 
weight, we might then indeed measure lifting power by the 
volume of the things lifted. But if the same volumes grow 
heavier, more lifting power is needed to lift them. And so if 
the same volumes or weights grow more valuable, more purchas- 
ing power is required to purchase them. Therefore it is equally 
absurd to measure general purchasing power only by either the 
volumes or weights of the thuigs purchasable. The general pur- 
chasing power of a sum of money can be measured only by the 
exchange-value of the things the money can purchase. The 
money has risen in exchange-value, for instance, either if it 
can purchase a greater number of things that have the same ex- 
change-value as before, or if it can purchase the same number of 
things that have a greater exchange- value than before. The ex- 
change-value of the things whose quantities purchasable are be- 
ing considered must also be taken into account. Therefore if we 
are measuring general purchasing power .only by the constancy or 
variation of the quantities purchasable, we must be careful that 
^ve be dealing with individual masses that have the same ex- 
change-value — over both the periods compared. 

Being thus identified with exchange-value in all other things, 
the conception of general purchasing power can give us no more 
trouble — and, of course, little aid. What has caused trouble is 
the fact that this idea has been introduced mto economics as a 
variant upon the idea of general exchange- value more intelligible 
than it and serviceable to help us in our conception of it, or to 
further us when it tails. Instead, by introducing comparison 



ITS FORMULATION 1)06 

between inessential attril)utes in things, this idea has only cansed 
confusion, and it is responsible for many of the aberrations in 

our subject. ^ 

III. 

§ 1. The measurement of the constancy or variation in the 
exchange-value of money by the numbers of equivalent mass- 
units is not, of course, confined to the case of two equally im- 
portant classes. It is as ap[)licable to any num})er of classes, 
with any amount of unevenness in their sizes, provided they 
satisfy the condition required for its application. Therefore 
we reach this general method of measurement : — When con- 
stant sums of money are expended at both periods, find in all 
the classes masses, to be used as units, that have the same ex- 
chanfjc-value {or money-value) over both the periods together ; and 
measure the constancy or variation in the exchange-value of money 
by the constancy or variation in the total number of such ma.ss- 
imits purchased at each period. 

If we have already executed the first part of this injunction, 
the true formula for the price variations, in these cases, is 

P.^- V^ + y," + V ^ + ... 

P, ■«/'-+^/' + V' + ' ^ ^ 

or a b c 

P a" "*" J'' "*" r'^ "*" 

P, a be ' ^^^ 

1 -I 1 1- 

^h Pi 1 2 

in which, as before, the doubly accented letters represent the 
numbers and the prices of such mass-units at each of the periods.^ 

* And yet, as the term contains reference to power, reference to which is lack- 
ing in the term " exchange-value," the term " purchasing power " ought to have 
been of assistance. That it has not been is probably clue to the limitation in the 
meaning of the term " to purchase " noticed in Note 4 in Chapt. V. Sect. III. § 8. 

^ The second of these formuhe is like a formula cited in Chapt. V. Sect. VI. 
§ 2 as a miscarried harmonic average of price variations. But there the formula 
was applied to any mass-units, while here it is applied only to special mass-units. 
The weighting here in this formula, viewed as a harmonic averaging of price 

a b 

variations, is according to —,,,'^-7;, , which agrees with what was there 

«i Pi 
said. Another interpretation of this weighting will be noticed presently (for 
a b 

— 7> = Xi", a~fi = y i"> ^tid SO on ) . 
fli Pi 

20 



306 THE METHOD FOR CONSTAxNTT SUMS 

The mass-units may be of any sizes, provided they be equiva- 
lent over both the periods. Thus in the examples above used 
in which the price of [A] rose by 50 per cent., we supposed A 
to be such that its price rose from 1.00 to 1.50, so that the 
geometric mean price of this and of every equivalent mass-unit 
was 1.2247. We might equally well have supposed its price to 
rise from .(30 to .90, so that its geometric mean price and that 
of every mass-unit equivalent to it would be .7348 ; or we 
might have supposed any other rise by 50 per cent, we pleased. 
For such clianges affect all the mass-units and all the prices in 
the same proportion, so that their numbers are all altered in the 
same proportion (the inverse of the preceding), and consequently 
the sums of their numbers ; wherefore the results are the same. 

§ 2. As .r/' = -^ and x^' = ~jj therefore ^,7 = ^ and 
"1 '■'■2 2 1 

■X." a" 
^, = '^,, ; and similarly in the other classes. Also, of course, 

xl' a^y 

the prices a^ and o.J' are in the same ratio as the prices a^ and 

a'' «., a" a 

«2 (the prices ordinarily quoted), so that —fj = — and — jj = — , 

and so on. Now then 



V^ + y/^-l- ^ 

x./'-\-ii./'+ 1 



+ 2//^ + 



(^>''v^ + ^'^t-+ ) 



i r..//^ij_ //^i 



(3) 



+ 2//^ + 



(■'■"^■"1+ ) 



and 

v^ + y/^^ 1 f,//V^ ,,yil^ . \ 

=,7'+W^(v'^; + y."4 + ). (4) 

The interpretation of these two equations, along with what we 
have discovered and presently shall discover about the third 
kind of mean and average, gives us the following proposition : — 
When constant sums of money are expended at both periods, 
the true result is always obtained (1) by the harmonic average 



ITS FORMUI.ATION 307 

of the i)ricc variations with wcif^hting accordinjj; to tiie numbers 
of the mass-units ecjuivalent over both the periods whieli the 
classes contain at the Jird period, or (2) bv the <iritlniiet'w aver- 
age of the price variations with weigliting acf^ording to the num- 
bers of such equivalent mass-units which the classes contain at 
the second period, or (8) in cases with only two e(iually impor- 
tant classes, by the f/eomefrie mean, and in most eases (provided 
there be no great irregularity in the sizes of the classes) veiy 
nearly by the geometric average with weighting according to the 
geometric means of the numbers of suc^h e(juivalent mass-units 
which the classes contain at each period ^ — ^vhich weighting is 
the same as the weighting of the classes according to the con- 
stant sizes of the classes at hofh periods.^ 

But the first two of these final solutions of our problem in 
regard to the proper averages of price variations (and their 
weightings) are of no practical utility, since they presuppose 
that we have performed the labor of finding the masses that are 
equivalent over both the periods together and the numbers of 
these contained in the classes at one of the jjeriods. If we 
have performed this lal)or, it would be easier to continue the 
operation in the way j)rescribed in the first of tiie above for- 
mulae. What we want is a method that will save us the great 
labor of such reductions. Here the geometric mean finds its 
recommendation in the cases where it is applicable ; and here 
the geometric average would be exactly what we want, were it 
only accurate in all cases. 

§ 3. It is, however, ]>ossible to make a formula that will in 
all cases yield the true result without requiring the finding of 
such equivalent mass-units — or rather a modification of the 
above formulae such as to contain within itself, so to speak, the 
method of selecting the mass-units, — a formula therefore, that 

- These general statements account for the particular statements made in 

Notes 4, 5, and 7 in Sect. I. of this Chapter. 

a / — 

* For Xi" = — 7^ and so on, and if the condition is given that V ai"a2" = 

a b 

V''/V'/V'= 't follows that l/aVW*"=T'<V7'' Vy7'y7' = i^^7/V = 

b 



V ai'a^", and so on ; wherefore T Xi'xo" : \^ yi"yn" : : : a : b : 



308 THE METHOD FOK CONSTANT SUMS 

is applicable to any mass-units we happen to use iu the various 
classes, requiring of us only that we know the prices of these 
mass-units, and the numbers of them purchased, at each period. 
Let us continue to use the doubly accented letters, x^", x.^', a^", 
a^', and the like, for the numbers and the prices of the mass- 
units which are equivalent over both the periods ; and let us, 
as generally, use the simple letters, x-^, x^, a^, a^, and the like, 
for the numbers and the prices of the ordinary mass-units which 
we find employed by merchants and referred to in statistical re- 
ports. Now it is possible to prove that 

a b c 



and 

— — a b c 

xya^a^ : y^^^^^ : zyr{(i ' '■ ZTf ' oTf '■ 777, - • 

"2 r2 12 

We may begin with the first two terms on each side of these 
proportions. As already explained, 

we have — = —j, and -^ = ^7, , 



«, « 



1 



"i<^2" J a ^\^2" 

whence a^ = —,7- and ^^ = -^ . 



Also by the hypothesis we have «/'«/' = j3/'i3^", 

/fO It (I "fj 1 1 

whence a/' = ^^^ and / V = ^ • 

Substituting these values in the preceding expressions, we have 



«A"A" „„,, . _ i>K< 



112 



and H„ = 



P2= ^^72 



"1 /^l 

And substituting these last values in the first expression to be 
proved, we have 

A ./TTTTTT, .. a . b 



-.547VA"/3/':y>^^<'<'::^:^ 



But x^a^ = a and y^^^ = b, and as s/^il'^-i^' = \/a/'a/', they 



ITS FORMULATION .30H 

may be eliniinatcd. Thus this ])ro|)(>rti()n is evident. Again, 
by a siinihir derivation from the known conditions, we obtain 

a = ^' • ' - and i - '^25_^_ 

and substitutinti^ these vahies in the second expression to be 
proved, we have 






'•2 /^2 "-2 P-z 

in which .c.//^ = ^ !"i<l .'/2i^2 = ^> '^"*^ *''^' remainders of the ex- 
pressions in the first lialf eliminate as before, so that this pro- 
portion also is evident. 

The same operation may be conducted upon the classes [A] 
and [C] (or ujion [B] and [C]), and so on through the whole 
list, with the same result. Q. E. D. 

Consequently 

^5,^+ yyi^ A + h^hTz + 

a b c 

^ <^ + /V^ + r7^ + 

a b c 

But we know by the second formula above given that the last 
half of this equation is the expression for the general price 
variation . Theref )re 

^2 ^ a;y«i «2 + y yt ^A + ^ynrz + 

Pi .ry^-\- nyjj, + sX /',?^ + ' '^^^ 

Here we have the formula desired, applicable to any mass-units 
and their prices and quantities, in all cases when constant sums 
happen to be spent on all the classes at both periods. 

This formula may be further metamorphosed. It may be so 
stated. 



310 thp: method foe constant sums 
a b c — 

p . - ^«lS + 15- ^^^2 + - ^*/V/'2 + 

f_2_'^l_ Jh^ /I 

R a ,— b >^^ c , — ' 

- ^'V'-2 + ^ <'^i/^2 + .T ^nr. + 

"2 /'2 /2 

Avhich reduces to 



^' = "^ — -^ — -^A — > (6) 

a 

'/?2 ' ^'r2 

and also to 



p. '^JI + ^JI + ^J^ 



p- .- ,- .- - (7) 



These are more curious than useful, except for theoretical pur- 
poses. The last leads ou through 



v47 + !'AJ~' + V.J^ + 



, .-„ J- 
P ~ 



^■''■Jf + ^AJt+^.'-.4 



to 

P2 ^ Sv^-Ci^^a + /32<yiy2 + r2^g2 + /gx 

which we recognize as a form of Scrope's method. It obviously 
is Scrope's method applied to the geometric means of the mass- 
quantities in every class at each period. Thus the method we 
have just elaborated turns out to be a new form of Scrope's 
method. 

§ 4. That this method does not violate any of our Proposi- 
tions concerning exchange-value in all other things goes almost 
without the need of notice. Taking any one of the formulae 
representing it (but most conveniently 6 and 8), we see plainly that 
it indicates constancy if no prices vary (Propositions XXVII. 
and XLIV.), and if all prices vary alike, it indicates the same 



ITS FOKMILATION ;5]1 

variation (Pro))(>siti()ns XYII. and XLV.), no matter what be 
the weights (])rovi(lcd, of course, they be tlie same at both 
periods) ; nor can it indicate constancy if there is onlv one |)ri(!e 
variation, or if all price variations are in one direction — and in 
the latter event it naturally cannot indicate a variation in the 
opposite direction (Propositions XX. and XXVIII.). But there 
is one Proposition by which it es])ecially needs to be tested. 
This is Proposition XXXVI. There is perfect evidence that if 
our method, ajijilied to all but a few classes, indicates a variation 

P, 

(or constancy) in the hsvel of prices so that p" = r, and if we find 

I 
that the remaining classes have all varied in price individually 

so that the price of each at the second period is r times its price 
at the first (or even if they have collectively done so), our method 
ought still, with these classes added, to indicate the same vari- 
ation. Let [C] embrace all the individuals left over. Formula 
(()) is here the most convenient. If our method, aj»plied to all 
classes except [C], indicates a variation in the general level of 
pri(^es from 1 to /•, we must have 

''^l^•'^g+ = '-(^J^^J^ )• 

But the price of [C] has varied so that — = r, and — = 

/'i r2 '■ 

Hence, extending our method to include [C], we should have 

+'=J,^)' 

Avhich is evidently true, because it reduces to 

+ C \^r. 






312 THE METHOD FOR CONSTANT SUMS 

Here we are supposing that z^^ = zj^ = c, in order that the 
class [C] may be properly subject to this method. If things 
were not so — if we had to distinguish c into c^ and c^, — the de- 
sired equality would not exist. Now that Proposition * does not 
require the omitted class or classes to be of the same money- 
value at both the periods. Still this method is not shown to be 
false because it would not be true if we applied it to conditions 
to which it does not pretend to be applicable. Confined to the 
conditions to which it is applicable^ it stands the test of that 
Proposition perfectly. ^ 

§ 5. That our formula (5), upon which depend the ones fol- 
lowing, is correctly derived, may be negatively shown in this 
way. If the mass-units have already been chosen that are 
equivalent over both the periods together, this formula reduces 
to formula (1), and so is correct. 

That this formula and the ones following it will reduce to 
formulae expressing the harmonic and the arithmetic averages, 
and in some cases the geometric mean, of the price variations, 
with the weightings above described (in § 2), is proved by the 
operation by which these formulae themselves were proved, since 
formula (2) to which they were proved to be equal is equal to 
formula (1). 

But with more than two classes, even if equally important, 
this formula will not universally reduce to the geometric aver- 
age of the price variations. 

Some special cases in which the geometric average of the 
price variations will regularly agree with this method and give 
the true result may easily be seen to be as follows : — (1) if all the 
classes are equally important and if there are 4, 8, 16, 32, 

64, such classes ; for then the common variation of two 

classes may be compared with the common variation of two 
other classes, and the common variation of all these four with 

* Nor Proposition XXXII., which comes into play if r = 1. 

5 A qualification, however, must be added. Above we used formula (6). If 
we used formulae (1), (2), (5) or (7), the same inability to satisfy that Proposition 
would remain. But if, instead, we used formulae (3), (4) or (8), (which contain 
nothing that shows their confinement to cases with constaut sums), and treated 
the additional mass-quantities as in any of them prescribed, this method would 
carry out that Proposition. 



r)p:VIATK)X OF THK (iKOMKTRU; AVEUAGE .'51. "J 

the common variation of fonr others, and so on ; — (2) if some of 

the chisses are doubly, quadrnj)ly, oetnply, larj>;e and in the 

preceding schema take the place of sets of 2, 4, H, classes ; 

(3) if beside such two or more classes all the rest are without 
price variation, or all have the same |)rice variation, and if the 
price variations of those classes, measured a|)art by themselves, 
show general constancy', or that same price variation ; for then 
constancy, or that variation, is the true result for all, and this 
will still be indicated both by the geometric average and by this 
method, as just shown, no matter (in the case of the geometric 
averaging) what be the weighting of tiiese additional classes. 
We shall also find a more general account of other more irreg- 
ular cases in which the geometric average may be exactly true. 
We must turn to the examination of the geometric average 
of price variations, this being generally so nearly true as to de- 
serve our attention bv itself. 



lY. 

§ 1. A couple of illustrative examples may be examined in 
full. AVe need to treat only of the next simplest cases. These 
are when we have three equally important classes to deal with, 
two of which vary alike, or — what is the same thing — two classes 
one of which is twice as important as the other. Here is room 
for two examples according as it is the more or the less impor- 
tant class that falls in compensation for the rise of the other. 
Ijct us begin with tiie former. In the usual form of starting 
M'ith prices at l.(H), our familiar example with A rising by 50 
per cent, may.be schematized as follows — for the harmonic 
price variati(Mis : 

I 100 A (7j. 1.00 200 B Oi 1.00 —300 I 100 for [A] 200 for [B], 

II 66| A (5 1.50 2331 R (;, .8o71( ^ f )— 300 | 100 for [A] 200for [B]; 

for the arithmetic price variations : 

1100 A (5 1.00 200 Br'r, 1.00— 300 I 100 for [A] 200 for [B], 
II 66| A @ 1.50 266| B fs .75— 333^ | 100 for [A] 200 for [B]; 



314 THE :method foe constant sums 

for tlie 2:eometrio price variations : 

I 100 A Or 1.00 200 B @ 1.00 —300 

II 6fi?Ar^l.50 244.948 B@ .8165 (==^/|") —311.614 

100 for [A] 200 for [B], 
100 for [A] 200 for [B]. 

In tlic first of these, in which the harmonic average of the 
price variations with the weights 1 for [A] and 2 for [B] in- 
dicates constancy, the compensation is (arithmetically) by eqnal 
quantities of the mass-units of [A] and of [B] that are equiva- 
lent at the first period — a compensation which, as before, is 
good at the first period only, and is too small for both the 
periods together. In the second, in which the arithmetic aver- 
age of the price variations with the same weighting indicates 
constancy, the compensation is by double gain on [B] over the 
loss on [A] in the same mass-units, this double gain being con- 
formable to the fact that the price of this mass-unit of [B] has 
sunk at the second period to half that of the mass-unit of [A] , so 
that this compensation is the compensation of the second period 
alone, — as may be perceived also by this arrangement : 



I 100 A («, 1.00 100 B' @, 2.00—200 
II 66|A@, 1.50 13.3i B^ @ 1.50— 200 



100 for [A] 200 for [B], 
100 for [A] 200 for [B], 



in which, A and B' now being mass-units that are equivalent at 
the second period, the compensation is by an equal number of 
such, and so is too great for both the periods together. Thus 
the argument from compensation by equal mass-quantities, ap- 
plied to the harmonic and arithmetic averages in these more 
complex cases, has the same reversed errors we have seen in the 
simple cases with two equally important classes. 

In the third schema, in which the geometric average of the 
price variations with the same weighting indicates constancy, 
there is no appearance of anything to recommend it, except the 
fact that its compensatory price and mass-quantity fall between 
the others. It has to be altered so that the mass-quantities 
compared be brought into better shape for the comparison. In 
it the price of B at the second period is the reciprocal of the 



DHVIATroX OF TTIK (JEOM KTIJK ' AVKHAOE 315 

geometric mean between the two prices of A, namely 1.2247. 
Tlie formulation for [B] may then he ch;ui<re(l into either 

I rxq B' (<i\ l.oO I 200 for [B], 
II 162.299 Ri (^, 1.2247 | 200 for [H], 
or 

I 162.299 B" (,i) 1.2247 I 200 for [B], 
II 200 B" Or, 1.00 | 200 for [B]. 

Still A\(' have not what will help us. Suppose then, making 
use of a hint given in C'ha])ter VII I. Section I. § (5, we take 
half of each of these alternative schematizations, as follows : 



I 665 B' dr. 1.50 + 81. 60 B" (x. 1.2247 
II 81.65 B' Or 1.2247 + 100 B" Or l.(»0 



200 for [B], 
200 for [B]. 



Here, however, there are two diU'erent mass-units of [B], the 
second being .8165 of the first. Therefore this arrangement 
does not in itself hel[) us. But we may combine the numbers 
given for the mas.s-units, and form a new single mass-unit, with 
the following numbers and prices : 



I 148.32 B'" @ 1..S4S46 
II 181.65 Bi"r'7', 1.10102 



200 for [B], 
200 for [B]. 



Here the ])eculiarity exists that the difference in the numbers 
of these mass-units is 33^, so that we have compensation offered 
by equal mas.s-quantities. Now if the geometric mean of the 
prices of this mass-unit were the same as the geometric mean of 
the prices of A, we should have exactly what we want ; for 
then this mass-unit of [B] would have the .same exchange-value 
over both the periods as the mass-unit of [A], and the gain of 
33^ of these mass-units of [B] would exactly equal and com- 
pensate the loss of 33 J of the mass-units of [A] . I^nfortu- 
nately this is not the case. By the nature of the formation of 
this mass-unit and its prices, its prices have 1.2247 for their 
arithmetic mean only, and their geometric mean is a trifle 
lower, being in effect 1.2185. Thus^these prices of B'" incline 
in the same direction as the prices in the harmonic extremes, 
also ofl'ering compensation by equal mass-quantities ; for the 
mean between those prices of B is far below that between the 
prices of A. We know that in those variations the price of 



316 THE METHOD FOR CONSTANT SUMS 

[B] has not fallen far enough. So, too, here we have an indi- 
cation that the price of [B] has not fallen quite far enough. 
There is compensation by gain of an equal number of mass-units 
slightly less valuable than the mass-units lost. Therefore money 
has slightly depreciated, and the price-level has risen somewhat. 
Another hint oifered in the same place in Chapter VIII. 
leads to the same conclusion. There we saw that it is possible 
to arrange things so that the prices of two similar mass-units of 
[B], falling as from 1.00 to .8165, together traverse a distance 
equal to that traversed by one A, namely when they fall from 
1.3623 to 1.1123. The schema for [B] would then be 

I 146.81 Biv @ 1.3623 | 200 for [B], 
II 179.80 Bi^' @ 1.1123 I 200 for [B]. 

Here the difference in the quantities is a gain of 32.99 such 
mass-units. The geometric mean of these prices is 1.23096. 
Thus while the compensatory quantities are slightly less, the ex- 
change-value of their units over both the periods is slightly 
greater. Which of these diiferences preponderates ? On [A] 
we lose 1.22474 x 33^ = 40.824 money-unit's worths (over 
both the periods together); on [B] we gain 1.23096 x 32.99 
= 40.609 similar money-unit's worths. So we gain less than 
we lose, and, as before, our money has depreciated, the fall in 
the price of [B] not being quite great enough. 

Now it is easy to place the prices of a mass-unit of [B] , fall- 
ing as from 1.00 to .8165, around 1.2247 as their geometric mean, 
and to find the numbers of these mass-units purchasable with 
200 money-units at each period. They are 



I 147.5579 B'' @ 1.3554 
II 180.7174 B'^ (a). 1.1067 



200 for [B], 
200 for [B]. 



But here the gain in such mass-units is 33.1595, or a trifle too 
little. This confirms the previous conclusion that the price of 
[B] has not fallen quite far enough ; for it indicates a slight 
diminution in the purchasing power of our money, since we are 
not permitted to gain quite so many equally valuable mass-units 
of [B] as we lose of [A] . 

What we want is to combine certain features which have so 



DEVIATION OF THE (iEOMETlMC AVEIJAGE ol7 

far not ctmic t()<i^('tlier, and winch will not come together so long 
as we snppose the price of [B] to fall as from 1.00 to .8105. 
AVe want a mass-nnit snch that its prices, falling, have 1.2247 
for their geometric mean, and such that the numbers of them 
purchasable with 200 money-units at the first and at the second 
|)('rio(l increase by '33^. If these two objects were obtainable 
with a fall of the price of [B] as from 1.00 to .SKio, the geo- 
metri(! average (jf price variations would be true in this and all 
such cases. But they are not, and we must seek for another fall 
of the price of [B] that Avill unite these requirements. It is 
not difficult to find the terms of this desired fall. The follow- 
ing is th(! schema for [B] : 



1 147.480987 B'^ O'l'^ 1.856107 
n 180.814333 IV (a, 1.106107 



200 for [B], 
200 for [BJ. 



Here we have a class, [B] , twice as large as the class [A] , and 
the price variations are such that for what we lose on the mass- 
([uantity of [A], the price of whose unit varies around 1.2247 as 
its ge(jmetric mean, we gain exactly as much on the mass-quantity 
of [B], the price of whose unit varies around the same figure, 
1.2247, also as its geometric mean, and whose units gained are 
equal in number to the other units lost.^ Therefore our gain is 
exactly equivalent to our loss over both the periods compared. 
Therefore our money has retained exactly the same purchasing 
})0wcr or exchange- value. But what is this jjrice variation ? 

rt is as from 1.00 to -^^^^= .815648 — a figure slightly 

lower, jis we have already found reason to expect, than the fig- 
ure required by the geometric method. Therefore if the price 
of B falls only from 1.00 to .8165, at which point the geometric 
average indicates constancy, it has not flillen quite far enough to 
counterbalance the rise of A from 1.00 to 1.50, so that the gen- 
eral level of prices has slightly risen, and the geometric average 

^ It may be noticed that the geometric mean between the numbers of these 
mass-units of [15] is 103. .3, which is just double the geometric mean between the 
numbers of the mass-units of [A], uamely 81.(35. This is im^argumenTwith Note 
2 above in Sect. II. It should also be noticed that the prices' oiTe very two such 
mass-units of [B] together traverse in the opposite direction exactly the same dis- 
tance as the price of the one mass-unit of [A]. 



318 THE METHOD FOR CONSTANT SUMS 

has, in this case, made an indication slightly below the truth. 

We should have been pleased if one of the familar and easily 
working averages, applied to the price variations with weight- 
ing according to the constant sums expended, in this case 1 for 
[A] and 2 for [B], would indicate constancy only when the 
price of [B] has fallen from 1.00 to .815648. Unfortunately 
none of the three do so.^ But of the three the geometric aver- 
age, with this weighting, comes the nearest to it. 

In default of a true average with the only weighting conve- 
niently indicated, we can, of course, resort to one of the formulae 
above discovered, preferably (5), and directly find the correct 
result without going through the long oj^eration above pursued, 
which has been done to illustrate the principle and to show the 
defect in the geometric average. Thus applied for the purpose 
of finding the proper variation of the price of B from 1.00 when 
A rises from 1.00 to 1.50 and at both periods 100 money-units 
are spent on [A] and 200 on [B] , in order to have constancy, 
the formula is 

100 X 1.224744 -\- 200v//^ 



= 1.00. 



200 - 
66f X 1.224744 -h--^v//9.. 

This reduces to j^^ -\- 0.2041 24v//92 = 1.00, which being worked 
out in the ordinary algebraical manner (with only positive terms) 
gives s/^2 = -903132, and ^9^= .815648. 

§ 2. The second example, in which the class with rising price 
is the more important, may be disposed of more briefly. Let 
us suppose that twice as much money is spent on [A] at each 

2 Or we should be almost as well pleased if we had some easily indicated sys- 
tem of weighting which with one of the familiar averages would give the true 
result. Now in the above example, when B falls from 1.00 to .815648, the weight- 
ing wanted with the geometric average is 1 for [A] and 1.9897 for [B]; with the 
harmonic, 1 for [A] and 1.4748 for [B]; with the arithmetic, 1 for [A] and 2.7122 
for [B] . Here the weighting for the geometric average is hopeless. That for the 
harmonic average is the ratio of the number of the (over both periods) equivalent 
mass-units of [A] to those of [B] purchased at the first period ; and that for the 
arithmetic average the ratio between them at the second period. This is in agree- 
ment with what is above shown in Sect. 11.^ g 2, which gives us a general principle 
for the weighting of these averages, but one, as there remarked, too laborious to 
be of utilitv. 



L- 



v^- 



DEVIATION Ol' TIIK (JKc ).MKTi;i< ' AVKItACJK .'] 1 !) 

poi'iod as on [B]. Now for ooiistancy to 1)0 indicated l)y tlio 
til !'('(! kinds of" averages each with the weights 2 for [.V] and 1 
for [Bj, the schemata ninst he as follows — for the hai'inonie 
averagre : 



I 200 A @ 1.00 100 B Oi], 1.00—300 
1 1 \?,\\\ A @ 1.50 16G| B @ .60— .300 

for the arithmetic average : 

I 200 A (a) 1.00 100 B Ol} 1.00—300 
II 133^ A @, 1.50 00 B (<3 

for the geometric average : 



200 for [A] 100 for [B], 
200 for [A] 100 for [B]; 



200 for [A] 100 for [BJ, 
200 for [A] ; 



1200 A(«)1.00 100 B(^ 1.00 —300 [ 200 for [A] 100 for [B], 

II 133^ A (nX 1.50 225 B (rt) .-1444 ( = |)— 358| | 200 for [A] 100 for [B]. 

In the first the compeu.satioii is still arithmetically by equal 
mass-quantities, measured in e([uivalents at the first period, and 
therefore ouly the proper compensation at the first period itself. 
In the second the recpiired change of j)riceof B is to zero, which 
makes A at the second period infinitely more valuable than B ; 
wherefore compensation is impossible. The third again needs 
to be restated. The restatement may be made, this time, by 
changing the mass-units of [A] . The price of A at the second 
period, 1.50, is the reciprocal of the geometric mean between 
the two prices of B. We may then have 



I 225 A' @ .4444 + 150 A" @ .6666 
II 150 A' @, .6666 -I- 100 A" Oi, 1.00 



200 for [A], 
200 for [A]; 



whence, by combining the ([uantities, and fi>rming a new mass 

unit : 

I 375 A"' Or, .5.333 ( = A) j 200 for [Al, 
II 250 A"' (a, .80 I 200 for [A]; 

Here the number of these mass-units lost is the .same as of the 
mass-units of [B] gained. But the prices are arithmetical, 
not geometrical, terms around .(j6GH, and their geometrical 
mean is lower, being .(3532. The conditions no longer resem- 
ble tho.se in the harmonic schema, in which the mean of the 
prices of the mass-unit which in equal numbers compensates for 
the gain of mass-units of [B] is itljove the mean of the prices 



320 THE METHOD FOR CONSTA^^T SUMS 

of the latter. The suggestion from this diiference is that the 
price of A has not risen quite high enough. We lose an equal 
number of mass-units that are not quite so valuable (over both 
the periods) as the mass-units gained. Therefore our money 
has augmented in purchasing power, and the price-level has 
fallen somewhat. The same suggestion is made when we find 
the mass-unit of [A] whose prices, rising in the same propor- 
tion as before, do vary around .666(3 as their geometrical mean, 

as follows : 

I 367.42349 K" @ .544331 j 200 for [A], 
II 244.95007 k^" @ .816497 ! 200 for [A], 

the number of these mass-units lost being smaller than that of 
the equally (over both the periods) valuable mass-units of [B] 
gained, so that money has slightly appreciated. To get the 
proper difference, 125, of mass-units of [A] whose prices are in 
the same variation around .6666 as their geometric mean, we 
find the following conditions required : 

I 368.942 A'^ @ .54209 j 200 for [A], 
II 243.941 .V @ .81987 \ 200 for [A]- 

Here at last we have the proper compensation, and the proper 

price variation of the class [A] .^ The proper price variation 

.81987 
is a rise as from 1.00 to \ , ^^^ = 1.5124. The same figure is 

.04209 

obtained more directly by use of the above-discovered formula 

(5). Applied to mass-units priced at 1.00 at the first period, 

this becomes 



200 K/a., -f- 100 X f 
x/Z + 225 X f 

«2 



= 1. 



which reduces to a.^ — 0.416666\/a2 =1, and works out, giving 
n/^2= 1.229804 and a.^= 1.5124. The figure is, this time, 
as we have found reason to expect, slightly higher than required 
by the geometric average. The geometric average is satisfied, 

^The geometric mean between the numbers of these mass-units of [A] is 
double the geometric mean between the numbers of the mass-units of [B] ; and 
the prices of every two such mass-units of [A] together traverse the same distance 
as the one mass-unit of [B]. Cf. above, Note 1. 



DKVIATION OK Tlir-; (iKOMETRIO AVKRACK 'V21 

and indicates c()nstan(!y, wlicn tlic price of A has risen only to 
!.")(), or l)ef'ore it has risen to its |)ro])er height, tliat is, when the 
conij)ensati()n ottered by its variation is not yet complete, and 
the general level of ])rices has fallen. Thus in this case the 
geometric average yields a result slightly above the truth/ 

^ ll. These examples, in conjunction with many others of a 
similar nature to be given later, ])ermit us to make an induction 
and to advance the following rule: — When the prlrcx fluit foil 
hclow the general (wenufe arc of preponderating ckitiHeH, the geo- 
vietric average, ivlth weighting according to the constant sums of 
money spent, yields a rej^iUt beloni the truth ; and when the prices 
th(d rise <d>ove the gener<d average are of prepondercding classes, 
it yields a result above the truth. 

We may form some idea of the probable amount of error in- 
curred by the geometric average, by measuring the errors in the 
above examples. In the first, the sums of the (over both 
periods) equivalent mass-units of [A] aud [B] purchasable 
with 100 and 200 money-units at the first and second periods 
were respectively 247..')579 and 247.3840 ; by dividing the 

P 

former by the latter we get p^ = 1.0007. Thus the geometric 

average, weighted according to the constant sums, indicated con- 
stancy where there really was a rise of 0.07 per cent. In the 
revision we saw that there would really be constancy if the price 
of B fell to .81564. If it did so, the geometric average would 
be 0.99932, indicating a fall of 0.068 per cent. In the second 
example, the sums of the (over both periods) equivalent mass- 
units of [A] aud [B] purchasable with 200 and with 100 
money-units were successively 46 7.42349 and 469. 95007, whence 

P 

—2 = 0.9946, indicating a fall of 0.54 per cent., although the 

geometric average, weighted as above described, indicated con- 
stancy. In the revision Ave found that to obtain constancy the 

* In the corrected example, with A rising from 1.0(1 to 1.5124, to indicate con- 
stancy, the weight wanted for [A], tliat for [B] being 1, with the geometric aver- 
age is 1.9602, with the harmonic 3.()89, with the arithmetic 1.084— the latter two 
being the ratios between the numbers of the equivalent mass-units at the first and 
at the second periods respectively. Cf. above. Note 2. 

21 



322 TH1-: MK'riioi) foi: constant sums 

price of A should rise to l.ol24. If it did so, the geometric 
average would be l.OOoo, indicating a rise of 0.55 per cent. 

These trifling errors, we may notice, are on larger price vari- 
ations than ordinarily occur in ])raetice between two successive 
periods. It is only with very large price variations that the 
deviations become appreciable. But also inequality in the sizes 
of the classes has something to do with the matter. If the pre- 
ponderating class varies much in price, while the smaller class 
varies little, the deviation may be appreciable, if the preponder- 
ance is moderate, but may be diminished almost to uothmg if 
the preponderance is excessive (the double excess having the 
eiFect of draggmg both the measurements after it till they almost 
coincide) — while, of course, a very slight preponderance also 
unites the results, approaching toward evenness of weighting. 
On the other hand, if the preponderating class varies little and 
the smaller class varies excessively,-^ the deviation may be great, 
and the geometric method becomes wholly untrustworthy. 

Examples of these rules may be given at random as follows : 

I 10 A @ 1.00 40 B r^, 1.00 I 10 for [A] 40 for [B], 
II 5 A @ 2.00 400 B (|, .10 | 10 for [A] 40 for [B]. 

The proper result iSp" = <).2()()5 ; the geometric method yields 

0.1821. 

I 10 A (a) 1.00 40 B @ 1.00 I 10 for [A] 40 for [B], 
II 5A@2.00 4000 B@. .01 | 10 for [A] 40 for [B]. 

P, 
The proper result iSp" = ().()4-15 ; the geometric method yields 

0.0288. 

I 10 A (a) 1.00 400 B (oj. l.(K) I 10 for [A] 400 for [B]^ 
II 5 A @, 2.00 40000 B (S; .01 | 10 for [A] 400 for [B]! 

P, 
The proper result is p'' = 0.0114 ; the geometric method yields 

1 
0.0110. 

^ Remember that tailing variations are larger than ri.sing variations the per- 
centages of whic-h, reckoned in the usual way, are equal. And this superiority 
rapidly increases with the amount of the variation. A fall from 1.00 to 0.99 is 
only slightly greater than a rise from 1 .00 to l.fll. A fall from 1.00 to 0..50 is twice 
as great as a rise from 1.00 to LfiO. A fall from 1.(10 to 0.01 is one hundred times 
as great as a rise fVoin l.on to l.liii. 



DIOVIATIOX OF TFfK <i KOMI-yPIMC A \- KIt.VG K 



I 100 A @ 1.00 10 liO^, 1.00 
II 50 A ©2.00 1000 Bra .01 



100 for [A] 10 for [B], 
100 for [A] K'for [B]. 



P, 
Tlic proper result isp"= ().8;}4'2 ; the geouietric method yields 
1 

Among uuiiiy classes, if some large ones rise in price while 
other large ones fall, and several small ones similarly offkit one 
another, and if the balances left over of large and small classes 
that vary in the same direction are small compared with the 
whole, the source of error again is small. 

For the errors are not cunudative. Tlint is to say, if the 
class [A] is twice as large as the class [B] , and the class [C] 
twice as large as the class [D] , and the prices of both [A] and 
[C] are the rising ones, so that in each comparison of [A] and 
[B] and of [C] and [D] the error may be, say, about half of 
one })er cent,, the erroi" Avill be no greater in the com|)arison of 
all four together. On the contrary, the error will be an aver- 
age somewhere between all the errors in the comparisons of an\- 
two unevenly matched pairs. And so, instead of accumulating, 
flie errors tend to neiitrah'ze one another. This follows by the 
principle of continuity from the two halves of the statement 
above advanced. For if in one [)air the prejionderating class is 
the rising one, the error is above the truth ; and if in another 
pair the preponderating class is the falling one, the error is be- 
low the truth. These two pairs of variations taking place to- 
gether, and being averaged together, the error must lie between 
them, and so must be diminished. It may perhaps disappear 
altogether. 

There is ako another way in which there luay be neutraliza- 
tion. This is through a series of successive periods. If during 
one comparison of two periods a large class rises in price 
through abnormal cau.ses, its variation will tend to cause error 
in the geometric average above the truth. If then in the com- 
parison of the next or some soon succeeding period its price falls 
back to its usual state, this reverse variation will tend to cause 
error in the geometric average below the truth. Hence in the 
series the later error will make up for the earlier. That this 



324 THE METHOD FOE C<)^■STA^'T SUMS 

kiud of compensation, if the reverse change is back to the same 
figure, the classes remaining of the same importance, is exact, 
will be shown presently. 

In the harmonic and arithmetic averages there is no snch 
tendency toward neutralization m successive periods. Their 
errors may accumulate indefinitely. 

V. 

§ 1 . What has been proved in the preceding Sections may be 
confirmed on a new line of reasoning, which also yields instruc- 
tion on points not yet proved. 

Suppose that in our simple hypothesis of a w^orld with money 
and two equally important commodities we have the following 
conditions, (the mass-units being chosen in the usual way, as 
equivalent at the first period), over three successive periods : 



I 100 A @ 1.00 100 B @ 1.00 

II 66t A @ 1.50 133J B @ .75 

III 100 A @ 1.00 100 B @ 1.00 



100 for [A] 100 for [B], 
100 for [A] 100 for [B], 
100 for [A] 100 for [B]. 



Here we have two sets of price variations, the first consisting of 
these, f and |^, the second of these, |- and |-. In addition we 
have the comparison between the prices at the first period and 
the prices at the third period. This comparison shows con- 
stancy ; for evidently, the state of things being the same at the 
third as at the first period, we have the same conditions as if 
the state of things at the first period had remained constant. 
We know, therefore, that after drawing the average of the first 
set of price variations and that of the second set, if we then 
attempt from these averages to find the average of the whole 
variation (or constancy) from the first to the third period — which 
is done by multiplying together the two single averages of the 
price variations, — the result should be unity, indicative of con- 
stancy. Here we have control over the result, and may use 
these conditions as a test case. 

It may be noticed at the outset that in all such cases (con- 
fined to three periods) the formulae above obtained satisfy this 
test. For the quantities and prices being exactly the same at 



THST CASKS ."Vio 

the tliird us at tlic first pcfiod, tlic first variation is in(licat(Ml, 
l)v tiirmula (•")), to he 

uiul the swoiul 

I', x-y^///, + !/yi-i^i% + 

, p. / p, p. \ 

Tlios(; are reciprocals; wherefore pM = p" . pM = 1.00. It is 

phiin that if the prices are irrc(/nl((rli/ diiferpnt at the third 
period, the direct comparison of the third with the first will not 
yield the same result as the indirect comparison of them through 
the intervening ])eriod. Wv shall not pause to consider this in- 
consistency here, as it will again call for attention in the com- 
plete method involving this partial method. We shall at pres- 
ent devote our attention to the certain test yielded by a revision 
of everything at a later period to what it was at the first. T^et 
us now see how the averages of the price variations stand 
this test. 

Averaging the ahovc sets of price variations harmonically, we 

2 2 8 

get, for the first, , ^ = 1.00, and for the second, g g = ^ 

= 0.888S ; and the product of these results indicates that be- 
tween the first and the third period the general level of prices 
has fallen by 1 1.1 1 per cent., which we know to be wrong — and 
to err by placing the level of prices too low. 

But suppose, after drawing the harmonic average of the first 
set, we draw the arithmetic average of the second, likewise with 
even weighting. We then have | (| -f- |) = 1.00 ; and each 
of these results indicating constancy, the whole result indicates 
constancy, which is right. 

In eflPect, the second price variations, from the harmonic terms 
on opposite sides of unity ti» luiity, are (simple) arithmetic vari- 
ations — that is, are the same as variations from unity to the 
opposite arithmetic terms. Thus at the second period the price 



326 THE .A[KTH<>D FOR COX8TAXT SUMS 

of I A is 1.00 aud the price of IJ B is 1.00, and at the third 
the prices of these parts of A and B are .66f and 1.33^ re- 
spectively, which are the arithmetic terms aronnd unity. Nat- 
urally, therefore, only the arithmetic average, Avith even weight- 
mg, will indicate constancy when prices vary in this way. 

Turning our attention to the quantities purchased, we see 
that with every two money-units we purchase at the first and 
third periods 1 A and 1 B, and at the second | A and | B. 
Thus at the second period, compared with the first, on the quan- 
tities of the first, the loss on A is one third and the gain on B 
one third, the gain equalling the loss. But at the third period, 
compared with the second, and with the gain and loss measured 
in the same way, that is, on the quantities at the earlier of the 
two periods compared as the unit wholes, the gain on | A is one 
half and the loss on |- B is one quarter, constituting the com- 
pensation by harmonic proportions. This agrees with the fact 
that the price variations in this case are arithmetic variations. 
But if the price variations really had been harmonic variations — 
namely, a fall of the price of f A from 1.00 to .66f (bringing 
the price of A back to 1.00) aud a rise of the price of | B from 
.33|- to .66f (carrying the price of B up to 1.50), so that the 
second harmonic average would indicate constancy, then not only 
would the level of prices at the third period be openly higher 
than at the first, but it is plain that the compensation offered 
w^ould be a gain on [A] , now the article falling in price, smaller 
than the loss on [B], now the rising article, so that, losing 
equally on [A] in the first variation, but gaining less in the 
second, we should lose on the whole. 
§ 2. Again if we had these conditions : 



1 100 A @ 1.00 100 B @ 1.00 

II 661 A @- 1.50 200 B @ .50 

III 100 A @ 1.00 100 B @ 1.00 



100 for [A] 100 for [B], 
100 for [A] 100 for [B], 
100 for [A] 100 for [B], 



and averaged the two sets of price variations arithmetically, with 
even weighting, the result for the first set would be 1.00, indi- 
cating constancy, and that for the second would be 1.33 J indi- 
cating a rise of 33^ per cent. But if we averaged the second 



I'KS'r CASKS .)J< 

s(>t luinnoiiieally, with vvvw wcitilitiii^i.-, we should have 1.00, 
and therefore 1.00 for the whole variation, eorreetly indicatinj^ 
coiistaiiev. Ill effect, the second set of price \ariatioiis are har- 
monic variations. ff" they really were arithmetic, such that the 
arithmetic average would indicate constancy, the price-level 
would really have fallen at the third j)eriod ; for though the 
|)rice of A would be back at 1 .00, the |)riee oi' ]] would be .()<)§ ; 
and our money, still s])ent eveiiK', would purchase more than 
before. 

^ .'). The order in which the two averages have been alter- 
nately applied might l)e inverted, and yet, although the particu- 
lar results would be diifereut in every instance, yet the wdiole 
results would be the same. Thus in the first example the arith- 
metic average of the first set of j)rice variations would indicate 
a rise of 12i per cent., and the harmonic average of the sec(md 
set would indicate a fall of 1 1.1 1 per cent.; but a fall from 1.125 
by 11.11 per cent, is a fall to 1.00. And in the last example 
the harmonic average of the first set of price \ariatioiis would 
indicate a fall of 25 ]ier cent., and the arithmetic average of the 
second set would indicate a rise of .'53^ per cent.; but the net 
r(\^ult of the two variations, |^ x | = 1 , indicates constancy. 

From these [)articular examj)les we get the following general 
rules, the universality of which is proved in Ap])endix B IV. 
§§4, 5, 7, that lohen the prices of two cqaa/fi/ inij>orfan( dasses- 
after vdrj/lnr/ at the second period revert at the third to irliat the if 
ice re before, flie continuous use of the harinonic average of tlie 
price variafions, with even weie/hting, gives a fined result known to 
/)e too foiv, and the continuous use of tlie arithmetic average of tlw 
price variations, loith even weighting, gives a fined result known to 
he too high ; hut the alternate use of these two averagcJi, similarli/ 
weighted, gives the final result known to be rigid. We have sutti- 
cient acquaintance with these averages to infer that these state- 
ments will extend to include all cases with any number of classes 
of any sizes, the only provisos being that constant sums be ex- 
pended on the classes at all the periods, and that the weighting 
l)e according to them. 

It must not be inferred, however, that an alternate use of the 



;32S THE METHOD FOR CONSTANT SUMS 

arithmetic and harmonic averages would at the end of every 
other period give the right result. Unless all the prices revert 
to what they were at the start, and the weighting is the same 
throughout, the correction offered by the error of the one aver- 
age for the error of the other has not full opportunity, or has 
too much opportunity, to work itself out. Yet this use of them 
would depart less from the truth than the continuous use of 
either of them alone. 

The inverse nature of these two averages shows the mistake 
of a procedure which has not infrequently been pursued. Sev- 
eral statisticians, taking some period as a base at which all 
prices are reduced to 1.00 or 100 and with which all subsequent 
periods are compared by use of the arithmetic average of the 
price deviations, with even weighting, have also worked back- 
ward to still earlier periods, comparing these with the same base, 
now later or second to them, still using the arithmetic average ; 
and have then strung out all the results in a single series. We 
now see that in order for the series to be of uniform nature, while 
the arithmetic average is used in comparing the later periods 
with the earlier base, the harmonic average should be used in 
comparing the earlier periods with the later base ; or conversely. 
Yet as these statisticians have used even weighting where it 
does not belong, it does not much matter what average they use. 

§ 4. Returning to our argument, we see that if the continuous 
use of the harmonic average over two periods gives a result too 
low, each single use of it gives a result too low ; wherefore in 
the first example, when the first use of the harmonic average in- 
dicated constancy, there was really a rise of prices. And if the 
continuous use of the arithmetic average over two periods gives 
a result too high, each single use of it gives a result too high ; 
wherefore in that example, when the second use of the arith- 
metic average indicated constancy, there was really a fall of 
prices. And reversely in the second example. That the two 
kinds of averages together give the right result at the end, in- 
fallibly shows that each one alone is wrong — and the one as 
much wrong on the one side as the other on the other. Now, to 
review, in our first example the indications are, 



'I'KST CASKS .*}29 

\>y tlie liarniniiic avorajjo : l)y tlie aritlniu'tic avcnif^i! : 

for the l.st variation 1.00, for tlie 1st variation 1.12."), 
" 2(1 " .S8SS, " 2(1 " 1.00, 

" whole " .8SS8; " wliole " 1.125. 

Here tlie rijj-ht result foi- the whole ('(miparisoii, 1.00, is the (/co- 
metric mean between these two wroiio- ivsiilts. Shnilarly in the 
HecimJ examj)le the rioht result, 1.00, is the geoiuetric mean 
between the other two, l.->->;l and .7"). 

Suppose now in the first example we avei"a>2;e the price vai"i- 
utions oc'oinetrietilly. The ocDiuetric mean of tlie tifst set of 

f;; ;', •> .... 

price variations is y = = l.OfiOO,' indicatiny- a rise 

' \ l> 4 I S 

of ().0() per cent. The ocoinetric mean of the second set is 



J 



■^ V = ' = 0.i»4'2S.- indicatino- a fall of ~).~'l per cent. 



The product of these two restdts is unity, metuiino- that the fall 
from 10().0«j by ^.T'i |)cr cent, is a fall to 1.00. In the second 
example, with the j)rices at the second period altered to the arith- 
metic terms around 1.00, the (>;cometric mean of the first set of 

I •> 
price variati(ms indicates w fall to " = 0.<S()()0 by l;5.40 per 

2 
cent., and the o-c'ometric mean of the second, a rise to , = 

1 3 

1 . 1 r)47 by 1.").47 per cent. The jiroduct of these again is unity, 
the rise fr(mi O.S()()0 by 15.47 per cent, being- a rise to 1.00.^ 
Thus in these cases — and the universality of this rule is proved 
in Appendix B I\\ § <> — tlir confiniioas me of the (/eoinetrie mean 
yive.s flw final result hiioira to he right. 

Let us, again, su[)])ose these conditions : 



100 for [A] 100 for [B], 
100 for [A] !••" for [^1. 
100 for [A] 100 for [B]. 



I 100 A {<<'. 1.00 100 B Oi) 1.00 
II m'i A (<e, l.:>0 l.')0 B Oi\ .t)()^ 
III 100 A (a) 1.00 ino IW" l.<»0 

1 This is the geometric iiu-uii l)et\vfeii 1.00, the harmonic mean of these vari- 
ations, ami 1.125, the arithmetic. Therefore, if the geometric mean of the price 
variations he the right one, the harmonic mean lias erred as much helow as the 
arithmetic has erred above, these proportions being measured geometrically. 

2 This is the geometric mean between 1.00, the arithmetic mean of these vari- 
ations, and 0.88SS, the harmonic. Therefore — as in the preceding note. 

'Here, too, the figures, 0.>S(i()0 and 1.1'>47, are geometric means, the first be- 
tween l.OOand 0."."), the second between 1 .00 and 1. :>;).?:>. 



P>oO THE METHOD FOR CONSTANT SU.MS 

The geometric mean of each of these sets of price variations, 
|, ^, and |, I, is unity; and the result for the whole is also 
constancy. In effect, not only the first price variations, but the 
second likewise are geometric variations ; for at the second 
period the price of f A is 1.00 and the price of 1| B is 1.00, 
and at the third period the prices of these parts of A and B are 
M^ and 1.50. Here f A has Mien just as B fell in the first 
set of variations, and 1 J B has risen just as A there rose. As 
regards the quantities purchased with every two money-units 
evenly distributed, at the second period compared with the first, 
and on the quantities of the first, they are | A and | B. Thus 
in the first quantity variations the loss is of J A and the gain 
of J B ; and in the second, reckoned in the same way, the gain 
is of J A and the loss of ^ B, — so that the greater gain and the 
smaller loss (as appears from this way of estimating them) fall 
alternately on the one and on the other. Therefore, as the ul- 
timate result of the two geometric averagings is correct, we have 
a clear advertisement either that the result of each of the geo- 
metric averagings is correct, or that the error in each is corrected 
by an equal error in the other. 

But in the preceding examjDles, in which we dealt with only 
two classes equally important, the geometric mean gave a result 
geometrically midway between the two erroneous results given 
by the harmonic and arithmetic averages, which averages cor- 
rected each other at the end of the second variation, showing 
that they diverged equally from the truth on each occasion. 
Therefore, in these cases, under these conditions, the geometric 
mean of each set of price variations, coinciding with the mean 
between their equal errors, was exactly correct. And therefore, 
in general, the geometric mean of price variations, whenever it is 
applicable, is correct. 

§ 5. So far we have dealt with only two equally important 
classes. Let us now briefly extend the investigation to cover 
uneven weighting. We may suppose the followuig conditions : 



1100 A @ 1.00 200 B@1.00 
11 66| A @ 1.50 233^ B @ .8571 ( = 
III 100 A @ 1.00 200 B @ 1.00 



100 for [A] 200 for [B], 
100 for [A] 200 for [B], 
100 for [AJ 200 for [B]. 



TKST CASKS 331 

Here tlie liarmonic jivcraoe iiulicates fur the first price variations 
constancy, for tlie second a fall to ().}>.'>3.') by 0^ per cent. The 
arithmetic average indicates for the lii'st price variations a rise 
to 1.0714 by 7.14 per cent., for the second c(»nstancy. The al- 
ternating nse of these two averages, in either order, gives the 
right final i-esult. The geometric average indicates for the first 
price variations a rise to i/^||^= l.O.'ViJ) by 3.29 ])er cent., for 
the second a fall as from 1.00 to i?'|| = 0.9681 by 3.19 per 
cent. The continnons use of the geometric average gives the 
right final result. But here the geometric results in the indi- 
vidual instancies no longer are geometric means between the 
other two. In fact, we now have no reason to expect even the 
right result to be the geometric mean between those other re- 
sults, since we are no\\^ dealing with higher powers. We do, 
however, know from our previous reasoning that the geometric^ 
uvevayc is not exactly right in its single uses. Hence the other 
alternative remains. We thus have demonstrative confirmation 
of our ])revious inductions ; for, as the final result is right, we 
see (1) that the error in the (/eoinetric averages of the price varia- 
tions must be above the trutli in the one instance and below it in 
the other — and the determining factors can oidy be found in the 
facts that in the one instance the larger class has fallen in price 
M'hile the smaller class has risen and in the other the larger has 
risen and the smaller fallen, although the ctmnection is not ap- 
parent ; and (2) that these opposite errors lanxt be (^geometrically) 
e<jual to each other. In the i)articular ease before us these con- 
clusions are borne out by a})plying any one of the above-dis- 
covered formulae. Any one of these gives for the first price 

. . . 307.634 

variations a rise to ^r7--~^,^,, = 1.033o bv 3.3') jier cent., and for 
29V.669 • ^ 

297.()()9 
the second a foil as from 1.00 to ,-.-,...= "••><>76 by 3.24 

])er cent. Thus the geometric average was slightly below the 
truth in the first instance (where the preponderating class fell in 
price), and in the second (where the })reponderating class rose) 
slightly above the truth (so that we may induct that such will 
be the connection between the causes and effects in all cases). 



332 THE METHOD FOR CONSTANT SUMS 

And the geometric indications are above and below the trutli in 
equal proportions; for 1.0335 : 1.0329 :: 0.9681 : 0.9676 (with 
as close approximation as may be obtained without using more 
decimals). These relations will be found to hold in all cases. * 

The general conclusion from this investigation is that in deal- 
ing with many classes in a long series of periods, where the 
many tendencies toward neutralization of the errors have room 
to play, it is extremely probable that the result given by the 
geometric average will never depart much from the true figure. 

But of course this conclusion is purely theoretical, since the 
condition presupposed, of constant sums (or sums constantly in 
the same proportion) being always expended on all the classes, 
is never fulfilled in practice. 

§ 6. The formula above discovered for finding the general 
price variation between two periods, in all cases when constant 
sums are expended at both periods, we have seen, satisfies this 
test whenever it is confined to a reversion of the state of things 
at a third period to what it was at a first period after a single 
deviation at a second period. Now suppose that the same state 
of things reverted to is at a still later period, after three or more 
variations through two or more intervening periods. Here we 
have a test for this formula applied in a series of measurements. 

Let us consider the next simplest case possible, illustrative of 
these complex cases — consistmg of two classes with three varia- 
tions extending over four periods. AVe may posit them in the 
universal, or algebraic form, as follows : 

a for [A] bfor[B], 

a for [A] bfor[B], 

a for [A] bfor[B], 

a for [A] bfor [B]. . 

* The argument here used concerning the geometric mean and average has been 
suggested by Westergaard's argument for the geometric average above examined 
near the end of Sect. VI. in Chap. V. But the details and the conclusion are 
ditt'erent. — That the geometric average works backwards as well as forwards, has 
also been noticed by Wicksell, B. 139, p. 8, but without pointing out the condi- 
tions and without employing this fact as an argument in favor of the geometric 
average in comparison with the other two. 



I — A @ fli 


^B@/S, 

Pi 


II - A @ a, 

"2 


^^ B @ ;3, 


a 

III — A @ «3 

«3 


\b@i3. 

Pa 


IV - A @ ttj 


\b@I3, 



TKST CASKS 333 

As the conditions are exacitly the same at the fourth j)eriod as 
at the first, it is evident tliat the exehan^e-vahie of money and 
the lev(!l of prices are then the same as at tlie first period, and 
ought to he so indicated in the series of measurements in wliieh 
the above foi-nuda is aj)plied to the three successive j)rice vari- 
ations — as, indeed, it is indicated by a direct com|)arison of the 
fotirtli witli the first ))eriod. For our present pur))ose formula 
(0) is the most serviceal)le. To find tiie result reached through 
the intermediate comparisons we have to multiply together the 
three applications of this formula, — one to each of the three sets 
of variations. To do this with the fornuda in its full form 
woidd be tedious. We may simplify the operation by substi- 
tuting the f(»llo\ving shortened symbols. J^'t a = .]—, b = I— , 

e<|ual the reciprocals, and may stand for them. Also he = <t, 
and </= <h The formula will now be 



j.^^(a/, + b<.)(a,. + b/)(^+^) 
a b\ /a \>\ " , ' 



p 



which, upon nudti])lying out, reducing and rearranging, be- 
comes 

(I (I e l> f <■ 

,-, a'-f-a-b ,-f-ab- -f-a-b , -|-ab- -fa-b" -fab- . + b* 

^A= '' " ^ '' '■ J _ /(,) 

'■ a+ab- , -i-a-b - -f-ab^y-f-a-b- -t-ab2--f-a-b -.-f-b^ 

. (I (I (■ (• t 

This can equal unity : (1) if « = <l, l> = c, and c = /, that is, if 
there are no })rice variations at all, or if all the price variations 
at any or every stage are in the same proportion ; (2) if a = h, 
d = e, and c =/, that is, if there are no price variations between 
the second and third periods (which virtually reduces the series 
to three ])erio(ls with two sets of variations), or if a = r}> and 
d = re, while c = /*, that is, if all the price variations are alike 



:>")4 THE METHOD FOR CONSTANT SUMS 

between the second and third periods ; (3) if a = b, that is, if the 
two classes are equally important. This last condition cannot 
be extended to cases with three or more classes indefinitely ; 
but it might be extended to ca.ses with 4, with 8, with 16, with 
32, classes ; or even with other numbers of variously im- 
portant classes if these arranged themselves suitably, in pairs or 
sets.'^ In other cases the formula does not equal unity unless it 
happens to contain elements that counterbalance one another in 
tending to deviations in opposite directions. 

Thus in this simplest example of the complex cases, used in a 
series of only four periods, the method fails to stand the test 
universally, even if everything is the same at the last period as 
at the first. 

§ 7. What is the reason for the failure of this method in a 
series of more than three periods ? — for the method has, appar- 
ently, been demonstrated to be true in every single comparison 
of two contiguous periods. The trouble is that the demonstra- 
tion has been directed at proving the formula to be true in every 
single com])aris(3n of two contiguous periods only in reference to 
each other and out of connection with any other periods. 

Over a series of years it would seem as if the perfectly true 
method should be modelled on what we have done in the case of 
a single measurement. That is, we should seek in all tlie classes, 
for service as units, masses that are equivalent to one another 
over all the periods compared, and should measure at every 
period the total numbers of these mass-units contained in all 
the classes at that period. Representing these numbers by 
x'", y'" , z'" , and so on, it is plain that if over four periods we 
had these results : 

^i" + Vi" + ^i" + to n terms = Q„ 



+ y-I" + V + *o '^ terms = Q^, 



•V + Vi" + V + ^^ ^ terms = Qg, 

•V" + Vi" + ^I" + to n terms = Q„ 

then the comparisons (according to formula 1 in Sect. III.) go- 
ing through this series, would yield 

■'' Cf. the conditions for the correctness of the geometric average above pointed 
out in Sect. III. ^5. 



'I'KSr CASES 330 

P, Q, Q, Q., Q,' 

just as if we compared the last period directly with tiic Hrst. 
And not only this, but also the comparison through the serial 
forms between any distant periods would yield the samt; re- 
sult as would a direct comparison between them, no matter 
what be the states of things at these periods or what the in- 
tervening variations, provided only the .same classes be used 
throughout. Thus, so emended, the method universally satisfies 
Professor Westergaard's general test. It is not difficult to show 
that tlie same results would be obtained by using, between any 
periods in an epoch of n/ jiericxls, this formula, applied to any 
mass-units and their prices, 



But in comparing every j)eriod first with the period preced- 
ing and then with the period following, if the state of things at 
the latter period be irregularly different from that of the former, 
the numbers which express at the second period the mass-units 
then contained that are equivalent over the first and second 
periods alone are not the same, or iii the same proportions, as 
the numbers which express at that period the mass-units then 
contained that are equivalent over the second and third periods 
alone. Therefore a direct comparison between the first and 
third periods in this same manner, namely by com])aring the 
numbers contained in the classes at each period of the masses 
tiiat are e(piivalent over the first and third periods alone, these 
numbers being again different, would not yield the same result 
as that yielded by the two former operations together. But if 
all prices are the same (or in the same proportion) at the third 
as at the first ])eriod, the mass-units used in both the compar- 
isons are the same ; and so, in such cases, there is necessary 

" The same satisfaction would be given by 

P, «2l/'^*i^"2^3 ^PiV^'yiViV^ ■■■ + 



Pi u^'^^ x^x^Xi +fiii/^yiy2i/:i + 

I)iit tiiis would not agree with the above in the individual result.- 



336 THE METHOD FOR CONSTANT SUMS 

agreement between their final result and the direct comparison 
between the first and third periods. But, again, even if all 
prices are the same at a fourth or later period as at the first, there 
has been a break in the continuity of the mass-units used ; and 
therefore the direct comparison, using one set of mass-units, and 
the indirect comparison, using two or more other sets of mass- 
units, will not necessarily coincide. 

The direct comparison, however, we must observe, will be no 
moreauthoritative than the indirect comparison through the suc- 
cessive comparisons of the intervening periods. The operation 
which uses the mass-units that are equivalent over all the periods 
compared, would emend both those operations. Yet again, that 
even this operation would be authoritative, does not appear. For 
this procedure is similar to that of using the geometric average of 
the price variations with the same weighting over all the periods, 
this weighting to be according to the geometric averages of the 
money-values at all the periods — a procedure against which ob- 
jection has already been raised.'^ 

Moreover, in practice it is impossible to employ this con- 
sistent method in a long series, especially as it would have to be 
revised in full every year upon the adding of a new period. We 
are practically confined to the course of comparmg every period 
with the periods immediately preceding and following, and of 
comparing distant periods through the mediation of such inter- 
vening measurements. It is desirable, then, to form an idea of 
the possible extent of the error in this procedure, and, if pos- 
sible, of the principles that govern it. 

§ 8. Suppose the class [B] is twice as large as the class [A], 
or, in other terms, a = 1 and b = 2. Then formula (9) becomes 

Again, suppose [B] is three times as large as [A] . Then 

P^ |_ ' d a b e c ./ 

_ ^ 

In riiapt. V. ? YT. near end 



3-;4-3- + 3-,' + 3- + 3^-h3-^+27 
a a e . c J 



+ 6-^+6-^ + 6^ 
a e / 



+ '-,+'i^'-c 



TKST CASES 337 

The divergent jKirts of the expressions are in the last three terms 

in tlie nnmerators and (U'noniinators. If [B] be four times 

larger than [A], the eommon terms in the brackets would be 

(t (I 

1 -f- 4 , -f 4 + + <!4, and the three terms in excess would 

ft (I 

{ d h c\ 
each luive 1 2 for their coeiheient. Let | -I- -f . ) be ren- 

V 'f '■ .t J 

resented by E', and ( / + / + r ) l\v K"- Then, with b=2 a, 

the first of the above formulte may be expressed in this simple 
manner, 

p^ ^ [41 + E' + E"] + E' 
P, ~ [4r-rE' + E"] +^" ' 

and with b = .") a, the other may be expressed thus, 

p^ - [91 -f E'> E"] + 2 E" ' 

and with b = 4 a, the formula would be 

P, _ [Uj^ + E' + E"] + 3 E' 
P, ~ [loT + E' +^:"] + 3 E'"' ' 

and so on. lu general, therefore, letting a = 1 and b = r, the 
comprehensive formula will be 



or 



\r' + l +E' +E"l+(7--l)E' 
1 [r' + ^ +E' +E"l+(r-l)E" 



p r' -f ^J + rE' + E" 



(11) 



P "~ 1 

• r- + + E' -f- 7-E" 
r 



(12) 



On the other hand, if [A] be the larger class, everything will 
work out merely to the inverse of the preceding, so that, with 
a = r' and b = 1 , the universal formula is 
22 



338 THE METHOD FOR CONSTAXT SUMS 

P ~ 1 

' /- + -, + /''E' + E" 



(13) 



P 

In these general formiilte, if /'(or /•' ) = 1, then ^^^ = 1.00 ; 

which is in agreement with the third condition above noticed 
for the reduction of formula (10) to unity. Again, as /• or /■', 

P 

increases toward infinity, the result, ^p^, approaches toward 

unity. This is in agreement with what was to be expected ; 

for as the one class infinitely predominates, the other sinks to 

nothingness, and the measurement is virtually by one class, but 

indirect (serial) comparisons with one class will always agree 

with direct comparisons with that class. These tAvo facts show 

that somewhere between equality in size of the two classes and 

the infinite predominance of the one, there is an inequality at 

which the error is at its maximum. Now from the way E' and 

E" are constructed, it is plain that they must always be about 

3. This being so, it is easy to find that the greatest possible 

error must always happen when the one class is about four times 

as large as the other. Roughly, also, it may be perceived, the 

greatest possible error will lie between one third and one fourth 

/ E" \ 

I 01' T^f > ''^^ the case may be 1 

unity. 

Consider the following example : 



E' / E' 
of the difference between ^,, ( or -^, , as the case may be ) and 



I 100 A @ 1.00 200 B@. 1.00 100 for [A] 200 for [B], 

n 661 A @ 1.50 260:1 B@ .75 100 for [A] 200 for [B], 

III 50 A @ 2.00 300 B@ .66s I 100 for [A] 200 for [B], 

IV 100 A @ 1.00 200 B@, 1.00 100 for [A] 200 for [B]. 

P 

To find vy as reached through the intervening comparisons, we 

need only calculate out E' and E" and apply them m formula 

(11) or (12). They are found to be 3.21(>3 and 3.2550 re- 

P 1 4 1 <S S ■-' 
spectively ; whence p^ = - \^.y_~ = 0.9972, indicating a fall of 



TEST CASES IVM 

O.li'S per cent., and committing- an error to tliat extent. If, 

keej)ing the same price variations, we supposed the mass- 

(juantities to be snch as to make the class [B] constantly three 

P 22.2378 
times larg-er than [A], we should have p* = 99~oT7[2 ~ 

P 

(».!)llfi47 ; if [B] were four times larger than [A], then p' = 

'^ 1 

„;-j J, ^~=. =: 0.99637 ; if [B] were five times larger than [A], 

P 44 5371 
then :pr^ = -rr-Frprrr, = 0.99648 ; — and thereafter the result will 
P, 44.6943 ' 

ris(! toward unity. Thus in this case the greatest possible error 

E' 

is about 0.3() per cent, behnv the truth. Here ^7, = 0.9879, 

which is 1.21 percent, below unity, and about 3^ times as large 
as the greatest possible error. 

If these examples were turned about, the same price varia- 
tions being kept, but the class [A] being supposed the con- 
stantly larger, the results would be the reciprocals of the pre- 

P P 

ceding. Thus with a = 2 b, p' = 1.0028 ; with a = :; b, p* = 

P P 

1 .00353 ; with a = 4 b, p' = 1.00363 ; with a = 5 b, p* = 

1 1 

1 .00352. The maximum error is about 0.36 per cent, above 

the truth. 

In the former of these examples ithe erroneous results were 
below the truth, following the gradual fall from the first period 
of the larger class ; and in the latter they were above the truth, 
following the gradual rise from the first period also of the larger 
class. If we inverted the intermediate periods, so that the 
third came second and the second third, the values of E' and E" 
would change places. Then in the former examples (with [B] 
larger) the erroneous results would be above the truth, follow- 
ing the gradual rise from the second period of the larger class ; 
and in the latter examples (with [A] larger) the erroneous re- 
sults would be below the truth, following the gradual fall from 
the second period still of the larger class. 



340 THE METHOD FOR CONf^TAXT SUMS 

If we make each class both rise and fall m price, we ^et the 
most satisfactory results by making them rise and fall together, 
as in the followiue: : 



I -100 A @ 1.00 200 B @ 1.00 

II 661 A @ 1.50 100 B @ 2.00 

III 150 A@ .661 2661 B@ .75 

IV 100 A (a). 1.00 200 B ® 1.00 



100 for [A] 200 for [B], 

100 for [A] 200 for [B], 

100 for [A] 200 for [B]. 

100 for [A] 200 for [B]. 

P 

Here E'= 3.0153 and E"= 3.0161, whence p*= 0.99994, in- 

1 
dicating a fall of 0.006 per cent., with a possibility of error 

E' 

through larger size of [B] showm by tv> =" 0.9997 not to ex- 
ceed 0.01 per cent, below the truth. The class [A] being the 
larger, the results would be inverted, — as also were the interven- 
ing periods alternated. And we get the least satisfactory results 
by making the classes rise and fall in price oppositely, — thus : 



I 100 A @ 1.00 200 B @ 1.00 

II 66| A @ 1.50 266f B @ .75 

III 150 A @ .661 100 B @ 2.00 

IV 100 A @ 1.00 200 B @ 1.00 



100 for [A] 200 for [B], 

100 for [A] 200 for [B]. 

100 for [A] 200 for [B], 

100 for [A] 200 for [B]; 

P 

for here E'= 3.5545 and E"= 3.7339, whence p* = 0.98S4, 

wrongly indicating a fall of 1.16 per cent. Witli [B] three 

P 

times as large as [A], p^ = 0.9851 ; with [B] four times as 

1 
P 
large as [A], ^^ = 0.9S-45 ; with [B] five times as large as 

P P 

[A], -p-^= 0.9846; and thereafter p* grows greater toward 

unity. Thus the error may, in this extravagant example, be as 
great as about 1 \ per cent. Again, if [A] Avere the larger class, 
the results would merely be inverted, — as also would be the 
case, again, if the second and third periods were reversed. 

In a series of five periods, the formulae would be still longer ; 
and longer still in a series of six periods, and so on. These be- 
come too complex to work out. AVe may rely on trial. And 
trial still shows but slight errors. 

In the above examples it is difficult to see a definite principle 



TKST (ASKS :'>41 

dctcniiiiiiiiti- which way the error shall p;o, or how far. A stato- 
mciit more than (tiicc made that //; (t series of four periods, if flie 
seeoiKl and third cIkiik/v phirc.s, the fin(d result is inverted, may be 
seen to be universal, by inspectinfi' the universal forniuhe. In 
a lon<>'er series this rule can a|)j)ly only to the alternation of 
periods e((ually distant from the first and the last. It is certain, 
tht'ii, that there is noti)iiiii' determining a continual error in one 
direction. And the errors, so small in the above extraordinary 
cases, will j)r()l)al)ly be but trifling in ordinary cases. In a 
general way there is some resemblance to the erroneousness al- 
ready found in the geometric averaging of the price variations, 
although the errors are probably smaller here. When many 
classes are dealt with, there is no reason to sujipose the errors 
Avill accumulate, but rather is there ])robability they will tend to 
neutralize one another. Also if the result happens to be above 
the truth at the fourth period, it is as likely as not to be below 
the truth at the fifth period, there being balancing from period 
to period. There is, then, extreme probability that in a long 
series this method will correct itself, and the results indicated 
bv it will alwavs hover in the neiii'hborhood of the truth. 

But the fact that this method shows error at all in a series 
of four or more i)eriods — not to mention inconsistency in a series 
of three periods — casts suspicion upon its perfect correctness 
even in a comparison of two periods. And yet the reason- 
ing by which it has been discovered has seemed to be sound. 
The method will later l)e involved in a complete method for all 
possible cases. Then will be the occasion to continue investi- 
gating the fault in it. 



CHAPTER XI. 

THE METHOD FOE CONSTANT MASS-QUANTITIES. 
I. 

§ 1 . As in the case of the precediDg argument, so the argu- 
ment from compensation by equal sums of money has been se^u 
to be applicable to all three averages not only when the subject 
of weighting is left out of account, but also when this is taken 
into consideration. The appearance of application solely to the 
arithmetic average is due to the selection of the mass-units in 
which the prices are reckoned ; for attention is often transferred 
from compensation by equal sums to compensation by equal 
prices. 

In a certain way by compensation by equal sums spent on 
constant mass-quantities ^ is meant a compensation by equal 
price variations ; but it is an equal variation in the price pri- 
marily of the whole quantity of every class that is purchased at 
each period. Thus if we represent the prices at the two periods 
of the constant whole quantity of [A] by a^ and a., , and those 
of the constant whole quantity of [B] by bj and bj , and con- 
fine our attention to these two classes, the position underlying 

P, a^b, 

stancy of the price-level, requiring that 2i,^ -)- b., = a^ -f- b^ , it 
requires (supposing A rises and B falls in price) that a., — a^ 

But the whole masses of [A] and [B] may be divided into 

^ As in the preceding Chapter, all that is said here is equally applicable to cases 
in which at both periods are purchased mass-quantities in the same proportion, 
■whether at the second period they be all larger or all smaller than at the first, pro- 
vided all the reasoning (and all the formulae later to be described) be applied only 
to the mass-quantities of the one or the other period — or only to what is common 
to both periods. Cf. also Appendix A, VII. § 5. 

342 



the present aro-ument postulates that ^ = — ^ ; and for con- 

^ ^ ^ R a, + b, ' 



(•()Mi*i;.\sAi'i(»N \\\ I'.c^iAL sr:Ms :]4l^ 

any iiuiubers ot" inass-iiiiits of various sizes, tlic prices of which 
are the ones cited. FTence for c )iistancv of the price level it is 
only the price variations of the agj^ret^ates of the niass-nnits 
that are re<piire<l to e<|ual each other, — not the |)arti<Milai" |)rice 
variations that happen to he reported. This is evident when 
we retleet that the one chiss may he nuich lar<;er and more im- 
portant than the other, in which case the price variation of any 
individual in the former is required by every one of the aver- 
ages to be smaller than that of an e(pial individual in the latter. 
For the present, however, we shall suppose ourselves to he deal- 
inu- with e((ually lari>-e or important classes, so as to have the 
cimvenience at tirst of dealing; with even weiirhtin"-. 

Now when the whole mass-quantities of every class purchased 
at each period are (Constant, but there are variations of prices, we 
have seen that the weights of the classes are different at the two 
periods, because the weights are according to th(» total exchange- 
values (or money-values) of the classes, and these must have 
varied under the conditions supposed. Hence, wanting to use 
even weighting, we have the option between three systems: (1) 
even weighting of the fird period, (2) even weighting of the 
■•second period, (.'>) even weighting of />()fli the |)eriods together. 
[f it hapj)ens that tlu; total money-values of the classes are 
equal at the first period, then the e([ual (•omj)ensation in their 
variations are from e([uality to ecpially distant opposite positions 
— which we know to be arithmetic variations. If it happens 
that the total money- values of the classes are equal at the sec- 
t)ud period, then the equal compensation in their variations are 
from equally distant opposite positions to equality — which we 
know to be harmonic variations. If it happens that the total 
money-values of two classes are alternately equal, that of the 
one at the first period being equal to that of the other at the 
second, and reversely, then the equal compensation in their 
variations are from opposite positions to reversely opposite posi- 
tions, traversing the same road in opposite directions, — which we 
know to be geometric variations. 

If, further, we divide the total masses of each class into ideal 
mass-units that are e(|uivalent at the first period, then if the 



344 THE METHOD FOR CONSTANT MAS8E8 

total money-values of the classes happen to be equal at the first 
period, and the variations of these are to the opposite arithmetic 
extremes, the variations of the prices of these mass-units will 
also be to the opposite arithmetic extremes. But if the total 
money-values happen to be equal at the second period and their 
variations are from the opposite arithmetic terms, although we 
have compensation by equal sums, we do not in this case have 
compensation by equal prices of these mass-units, which have 
varied to the opposite harmonic extremes. And similarly there 
is no equal compensation in the prices of these mass-units when 
there may be such compensation in the sums, if the total money- 
values happen to be equal over both the periods together. Hence 
in these cases, although the compensation by equal sums really 
exists as well as in the first case, yet the compensation by equal 
variations of the prices that happen to be cited has disappeared. 
This is why the argument under consideration has seemed to be 
an argument peculiarly in favor of the arithmetic averaging of 
price variations ; for statisticians, as we know, have formed the 
habit of using the variations of prices equal at the first period. 
But if we used mass-units equivalent at the second period, or 
over both the periods together, we should get compensation by 
equal prices (of such mass-units) even when the price variations 
are the harmonic or the geometric. We need now to review 
these possible positions. 

§ 2. Using mass-units that are equivalent at the first period, 
we may construct the following schemata illustrative of the con- 
ditions when there is compensation by equality in the variations 
of the total money-values of the total mass-quantities — in the 
cases, to begin with, of two classes siq3posed to be equally im- 
portant at some period or periods. The only changes in the 
construction of these schemata from those in the last Chapter is 
the omission of the sums of the mass-units at each period, and 
instead the insertion, at the extreme right, of the total sums of 
money expended at each period on both the classes. Thus on 
mass-units equivalent at the first period we have compensation 
by equal sums when the price variations are to the simple arith- 
metic extremes, as follows : 



COMI'KNSATION IJY EQIAL SUMS 34") 

I 100 A (», 1.00 100 K Of. 1.00 I 100 for [A] 100 for [R] — 200, 
II 100 A Oi) 1.50 100 B Ot .50 | 150 for [A] 50 for [P,] — 200; 

when the price variations are to the sinipU' harmonic extremes : 

I 100 A ^1.00 200 n (« 1.00 I 100 for [A] 200 for [B] — 300, 

II 100 A @ 1.50 200 I? (m jr, \ 150 for [A] 150 for [B] — 300; 

wlieii the price variations are to the simple geometric extremes : 

I 100 A (5j, 1.00 150 J5 (« 1.00 I 100 for [A] 150 for [B] — 250, 
II 100 A C'i\ 1.50 150 r. Oi .(iOn I 1.50 for [A] KM) for [B]— 250.2 

Here it is only in the first seliema that we have compensation 
by e(|[ual prices; bnt in all of them we have compensation by 
equal sums. We find also here the [)eculiarities with which we 
are familiar. In the fii'st the sums are equal at the first period, 
and the compensation by equal sums is away from this equal 
condition. In the second the sums are equal at the second 
period, and the compensation is by ecpial variations from the 
first condition to this e([ual second condition. In the third the 
ecpial sums alternate and change places, traversing not only 
e(iual distances, but the same road. 

That these peculiarities are unixersal may likewise be proved 
bv tlie formula discovered in § T) of Section II. in Chapter IX. 
In that formula the mass-units were supposed to be equivalent at 
the first i)eriod, wherefore it is applicable here. Letting the 
price of A always be supposed to rise from 1.00 tt) «./, we know 
that if the price of B falls from 1.00 to the opposite arithmetic 

extreme it falls to 2 — '/...', if to the liarmonic, to , y" — ^> ii 

- (f-.^ — 1 

to the geometric, to , . Supplying these values of ,9./ in the 

formula 

< - 1 

■" =r'->7' 

2 We might also here make the total sums always add up to tlie same figures, 
as follows for the second and third : 

I 6GS A @ 1.00 13:3^ B (a\ 1.00 I m'i for [A] Vm for [H]— 200, 
II (ifis A @ 1.50 133* B frtt .75 | 1(M» for [A] 100 for [B]— 200; 

I 80 A @ 1.00 120 B @ l.OU I 8(1 for [A] 120 for [B]— 200, 

II 80 A @ 1.50 120 B @ .(i()i^ I 120 for [A] 80 for [B]— 200. 

But here again the diiferenees in the sums are diiferent. Cf. Note 2 in Sect. I. of 
the last Chapter. 



346 THE METHOD FOR CONSTANT MASSES 

we find, when tlie price variations are to the arithmetic ex- 
tremes, 

.'/" = 1 ; 

when they are to the harmonic extremes, 

y" = -la,^ -\; 
when they are to the geometric extremes, 

/' = <• 

The first of these expressions means that when the price vari- 
ations are to the opposite arithmetic extremes, in order to have 
compensation by equal sums, we must purchase au equal number 
of mass-units of [A] and [B] ; but these mass-units are, by 
hj^othesis, equivalent at the first period ; therefore we must 
spend equal sums on [A] and [B], or spend our total sum 
evenly, at the first period. This, moreover, is evident, because 
the prices of these mass-units being supposed to vary to the op- 
posite arithmetic extremes, it is only by these mass-units being 
purchased in equal numbers that the total sums will vary at the 
second period to the opposite arithmetic extremes. The second 
of these expressions means that when the price variations are to 
the opposite harmonic extremes, in order to have compensation 
by equal sums, we must purchase for every mass-unit of [A] 
2 «./ — 1 mass-units of [B] equivalent at the first period, that 
is, for every money-unit spent on [A] at the first period we 
must spend this number of money-units on [B] at that period ; 
but at the second period we must spend a.^ on this 1 A and on 

these (2 «./ — 1) B's we must spend (2 «./ — 1) • . -, ^^ ,' _ -^ = «,, , 

that is, we must spend equal sums on [A] and [B] , or spend 
our total sum evenly, at the second period. The third means 
that when the price variations are to the opposite geometric ex- 
tremes, in order to have compensation by equal sums, we must 
purchase at the first period for every 1 A a.^ B's, and these mass- 
uuits being equivalent at the first, period, we must spend our 
monev in these proportions at the first period, that is, a..^ money- 
units on [B] fjr every one money-\mit on [A] ; but at the 



C'OMI'KNSATION I!Y IX^TAL SUMS 347 

second period to j)iir(;hast' the 1 A \vc liavc to spend n.,' tuoiiev- 

units, and to i^iirchase the <i.! \V)^ we have to spend a..' ■ , = 1 

- a.' 

money-unit, that is, the sinus must at the second period be the 

reverse of what they were at tlic; fii-st.' 

§ 3. If we use different mass-units, the formuUu and the re- 
sults for //" (the ratio between the mass-units in the classes) 
will be different ; but the results al)<)ve obtained for the total 
sums to be spent on [A] and [B] at each ])eriod will not be af- 
fected. Therefore these results are universal. 

We may, then, ])ass (juicikly over the schemata for the condi- 
tions when we use mass-units ecjuivalent at the second and over 
both periods. These may be easily obtained from the preced- 
ing by rearrangement of the mass-units, and their prices, of 
[B] . Thus with mass-units equivalent at the srcond period, we 
have, for the arithmetic price variations : 



[ 100 A@, 1.00 33i B^@ 3.00 
11 100 A @ 1.50 33J W @ 1.50 

for the harmonic price variations 

I 100 A @ 1.00 100 B^ @ 2.00 
II 100 A @ 1.50 100 B^ @ 1.50 

for the geometric price variations 

I 100 A @ 1.00 66| B^ @ 2.25 

II 100 A @ 1.50 66;| W @, 1.50 



100 for [A] 100 for [B]— 200, 
1 50 for [ A ] 50 f or [ B] —200 ; 



100 for [A] 200 for [B]— 300, 
1 50 for [ A ] 1 50 for [ B] — 300 ; 



100 for [A] 150 for [E5]— 250, 
150 for [A] 100 for [B]— 250, 



Here the particular and total sums are the same as in the pre- 
ceding examples, only the numbers of the mass-units of [B] 
being changed. It is now only in the harmonic price variations 
that compensation takes place by equal prices. 

With mass-units equivalent over hath periods we have for the 
arithmetic price variations : 



I 100 A @ 1.00 57.73 W @ 1.7320 
II 100 A @ 1.50 57.73 B'' @ .8660 



100 for [A] 100 for [B]— 200, 
150 for [A] 50 for [B]— 200; 



' In this case, if out of 2 money-units we spend Uo' times more on [B] than ou 

2 -02' 

[A] at the first period, we must then spend y^ on [A] and 7^^ on [B]; 

and at the second period the reverse. These figures, as before, at arithmetic 

extremes around 1, are the harmonic means between 1 and — , (/. e., ^i^) and 1 

and a^ respectively. Cf. Note 3 in Section I. of the last chapter. 



348 THE METHOD FOR CONSTANT MASSES 



for the harmonic price variations : 

I 100 A @ 1.00 141.42 B'^ @ 1.4142 
II 100 A @ 1.50 141 .42 B'' (n 1.0606 

for the geometric price variations : 



100 for. [A] 200 for [B]— 300, 
150 for [A] 150 for [B]— 300; 



I 100 A @ 1.00 100 B'^ @ 1.50 
II 100 A @, 1.50 100 B^^@ 1.00 



100 for [A] 150 for [B]— 250, 
150 for [A] 100 for [B]— 250. 



And here, too, the particular and total sums are the same as be- 
fore, only the numbers of the mass-units, and their prices, of 
[B] being changed. And it is now only in the geometric price 
variations that compensation takes place by equal prices, 

II. 

§ 1. Now the argument from compensation by equal sums 
must claim that in every one of the three price variations above 
thrice schematized the exchange-value of money, or the general 
level of prices, remains constant, because this comj)ensation ex- 
ists in them all. But, in each of the three sets, in the first ex- 
amples, which always illustrate the arithmetic price variations, 
the arithmetic average indicates constancy only with even weight- 
ing — and this is the weighting only of the first period. In the 
second examples, always illustrative of the harmonic price varia- 
tions, the harmonic average indicates constancy only with even 
weighting — and this is the weighting only of the second period. 
In the third examples, illustrative of the geometric price varia- 
tions, the geometric average indicates constancy only with even 
weighting — and this is the weighting only of both the periods 
together. And these conditions we know, by means of the 
formula above employed, to be universal. 

We find also that if we use the arithmetic average upon every 
single one of these schemata with the weighting of the first 
period, which is always uneven in the second and third ex- 
amples in each set, we always get indication of constancy ; that 
if we use the harmonic average upon every single one of these 
schemata with the weighting of the second period, which is al- 
ways uneven in the first and third examples, we always get in- 
dication of constancy ; that if we use the geometric average upon 



c<)i\<'ii)EN'('K OF 'I'liK A vki;.\(;ks ,'J4f) 

every single one of these s(!lieinat!i with tlic wcii^litino- of />o/A 
the periods (geometrically euleiilated, aeeonling to the square 
roots of the ])rodiicts of the total iiioiiey-values at each period, 
as explained in Chapter IV. Section V. § 0), we always g-et, at 
least vei'v nearly, indi(!ation of eonstaney. 

From these partiinilar facts, the first batch of which have al- 
ready been universalized, are to be derived two important infer- 
ences, the first of which can be directly universalized, and the 
second will be universalized presently so far as it admits of uni- 
versalization. 

§ 2. The first of these is this : — Constant mass-(iuantities be- 
ing purchased at each period, we can make use of the argument 
from compensation by ecjual sums in fiivor of the aritJniicfic 
average only if we confine ourselves to using this average Avith 
the weighting of the first period ; we can make use of this same 
argument in fixvor of the li<(rinonir average if we confine our- 
selves to using it with the weighting of the second period ; we 
can make use of the same argument in favor of the (/count rlc 
average (with a modification) if we use it with the weighting of 
both the periods together. The argument is as good for one of 
the averages as for another — imqualifiedly in the case of the first 
two, — provided each be used with a special weighting ; and for 
them with other weightings the argument has no force whatever. 

Thus we see that on the assum])tion that this argument is 
correct, and on the supposition that it has application (that con- 
stant mass-quantities are purchased at each period), each of the 
three averages has for its oavu proper weighting each of the three 
kinds of weighting jiossible — on the supposition of constant 
mass-quantities and varying total money-values of the classes. 
In these cases the proper weie/Jdinf/ to 7(se ivdh the aritlunctic aver- 
age of price variations is the weiglding of the first period ; tJie 
proper weighting to use with the harmonic average of price varia- 
tions is the weighting of the second period ; the proper iveighting 
to use fvith the geometric average of price variations is the weight- 
ing of both the periods (the weight for every class being the 
geometric mean between its two weights, one at each period). 

Here we might be inclined to argue in favor of the geometric 



350 THE METHOD FOR CONSTANT MASSES 

average on the ground that the weighting it requires is alone 
the right one. But we are deterred from so doing by the second 
inference. 

§ 3. This second inference is that all three, averages, each ivith 
Us oion proper weighting, applied to the same cases with constant 
mass-quantities sometimes yield the same results, and ahnays the 
first two i/ield the same results, and in most (^ordinary) cases the 
third depjart.s hut slightly from them. 

That, when the mass-quantities are constant, the arithmetic 
average of the price variations with weighting according to the 
sizes of the classes at the first period and the harmonic average 
of theni with ^veighting according to the sizes of the classes at 
the second period are identically the same universally is demon- 
strated in Appendix A, TV. § 3. For they both, being averages 
of price variations, reduce to a comparison of the same price 
averages (arithmetic, therefore in the same ratio as the price 
totals). Thus, condensing that demonstration, we have 

p;-.™,+^/A + (™x+^"^x+ ) (!> 



.aY/2 + y^^ + 



(3) 



This is nothing else than Scrope's method of measuring varia- 
tions in the level of prices, rightly confined to cases when con- 
stant mass-quantities are supposed to be given. A shorter, but 
less perspicuous, formula for this method is 

P. - a, +'b, + • ^ ^ 

Thus in all the schemata employed in this Chapter, to find the 
constancy or variation of the general level or average of prices 
(or the average variation of prices), we have only to divide the 
total sum supposed to be spent on all the classes at the second 



<(H.\("ii)i:.\('K OF THK avki!A(;ks 351 

period by tlic total siiin s|)ciit on tlicin at the first period. The 
more laborious operations of avera<2;ino; the priee variations arith- 
luetically with the wei<j,lits of the first period, or of averaging 
them harmonically with tlu> weights of the second period, may be 
dispensed with. 

That, on the same assnmption of constancy in the mass-(juan- 
tities purchased at both periods, with additional confinement to 
the supposition that we are dealing with the variations of only 
tiro classes, which, furthermore, are such that the geometric 
means of their two weights (at the two periods) are equal, so 
that we can use the geometric average of their price variations 
with even weighting, the geometric average with this weighting, 
that is, the geometric mean, universally yields exactly the same 
results as the other two averages, applied to these same cases, 
each with its proper weighting, is demonstrated in Appendix A, 
Vr. § 7. There also, in § J), it is shown that in all cases, with 
any number of classes, with variously uneven weighting, with 
all the moderate irregularities in the sizes of the classes such as 
are likely to exist in our statistical lists, the geometric average 
with its j)roper weighting always yields results very nearly the 
same as the single results yielded by the other two averages, 
each with its proper weighting. Of these more complex cases 
we shall treat in a later Section. Here we may continue to con- 
fine our attention mostly to the simple cases where all three of 
the averages, or rather means, each with its jiroper weighting, 
coincide. 

.^ 4. A pretty illustration of this coincidence may be offered 
in the following, which provides us with what may be called an 
argument by transposition of jirices. Suppose we have at each 
period 1 A and 1 B, and it happens that their prices become 
transposed, that is, that the price of A at the second period is 
the same as B's at the first and that of B at the second the same 
as A's at the first, so that o.^ = fi^ and ^^ = a^. Then also the 
importance of these articles is transposed — and if there be the 
same number of them in their classes, the sizes of the classes are 
to each other as these prices, and are transposed (as in the third 
s<;hema in the last set above). Under these variations it is per- 



352 THE METHOD FOE C0NSTA:NT MASSES 

fectly evident that the exchange-value of money suifers no varia- 
tion. Now this constancy of the exchange-value of money, as 
shown in the level of prices, is indicated by the arithmetic 
average of these price variations -with the weighting of the first 

period, thus ^ 1- — - , and by the harmonic average of them 

«i + /^i 

with the weighting of the second period, thus ^— — ^-^ , be- 



'«2 /^2 



cause each of these reduces to -^ ^ , which, on account of the 

«i + l-i, ■ 

equality between a^ and ,9^ and between ji^ and a^, is equal to 
1.00 ; and also by the geometric average with the weighting of 
both the periods, which is even, since "^ o.^o.,^ = v^/^j^J^ > because 

the expression ^ —. •;5? is equal to 1.00 also on account of the 

equality between those terms. Hence the argument based upon 
this perfect evidence is equally good for either of these averages, 
each with its own proper weighting.^ 

§ 5. On account of this coincidence, when the physical foun- 
dations are the same at both periods, it is futile to argue for any 
one average alone, or for any one weighting alone, in preference 
to any other. As regards mere convenience, Scrope's method is 
the most serviceable of all, in these cases ; and this represents 
no one of the three averages of the price variations more than 

^ This example may be widened, showing the same identity between the aver- 
ages differently weighted. Let the point Oi somewhere divide the line AB at the 
A O^ O2 B 

first period, and the point O2 somewhere divide it at the second period. Then the 
segment AO^ has become AO2, and the segment OjB has become O2B, but 
these variations of the segments have not affected the whole line ; wherefore the 
average of the variations evidently is unity, indicating constancy. Now what- 
ever be the variations, provided neither segment be zero in length at either 
period, constancy is shown by the arithmetic average of them with the weight- 
ing of the first period, and by the harmonic average of them with the weight- 
ing of the second period ; and provided they be such that AOi = O2B and 
A02 = 0iB, also the geometric mean (with even weighting) indicates constancy 
(and in other cases the geometric average with the weighting of both periods, 
provided the weight of the one be not more than two or three times larger than 
that of the other, gives a result very nearly equal to unity). 



C'OINCIDENf'E OF THE AVP^RAGES 353 

another, except in complex cases, when the geometric average 
deviates. There is little to say in favor of the arithmetic aver- 
age, or in favor of the weighting of the first period, each by 
itself; but if together they give the right result, we cannot com- 
plain. Similarly with the harmonic average and the weighting 
of the second period. The average and the weighting that have 
most reason in their favor singly are the geometric and the 
weighting of both the periods. Yet we find that these give no 
better results than the others properly combined, — and in com- 
plex cases we shall find that their results are not so good as 
those of the others. The greatest errors arise from using an 
average with weighting not suitable to it — the arithmetic with 
the weighting of the second or of both jieriods, the harmonic 
with the weighting of the first or of both periods, the geometric 
with the weighting either of the first or of the second period 
alone. 

In reality, however, not one, but all the averages and weight- 
ings are used. In Chapter IV. (Sec. V. § 4) we found it 
absurd to use the weighting of either period alone. But now, 
in using Scrope's method, we are not using the weighting uf 
either period alone. For we are using the weighting of the first 
period alone only with the arithmetic average ; but equally well 
are we using the weighting of the second period — alone with the 
harmonic average. The one true method which combines both 
these averages, combines both these weightings.^ And in the 
special cases where the result given by it is necessarily given 
also by the geometric mean, this is with the use of the combined 
weightings of both the periods.^ 

2 In Note 12 in- that Chapter and Section it was stated that a variation is not 
properly a variation of the individuals existing at either period alone, but it is 
from the individuals at the first period to the individuals at the second. We have 
now found that in averaging from individuals at the first period the proper aver- 
age to use is the arithmetic with the weighting of the first period ; and that in 
averaging to individuals at the second period the proper average to use is the har- 
monic with the weighting of the second period. Thus we have perfect harmony 
throughout. 

•'' Students of German philosophy will notice a peculiar resemblance between 
the three averages and the three categories in each of Kant's four divisions, and 
the three terms in the trichotomy of Hegel. For the arithmetic and harmonic 
averages are opposed to each other (although with likewise opposite weighting 

23 



354 THE METHOD FOR CONSTANT MASSES 

§ 6. Consequently in his dispute with Laspeyres, in which 
the price of one article was supposed to rise from 1.00 to 2.00 
and that of another to fall from 1.00 to .50, Jevons would have 
been right in considering the exchange-value of money con- 
stant, as indicated by the geometric average with even weight- 
ing, if he could have shown, or had added as part of the sup- 
position, that the two classes were equally important over both 
the periods together. But m this case the other two averages, 
each with its proper weighting, would make the same indication. 
On the other hand Laspeyres would have been right in using the 
arithmetic average with even weighting, and in concluding that 
the price level had risen by 25 per cent., had he been careful to 
specify that he was dealing with classes equally important at 
the first period, and only with such. But in this case the har- 
monic average with its proper weighting would make the same 
indication, while the geometric average with its proper weight- 
ing, would diverge only by indicating a rise of 26 per cent. 
Again, Jevons, in suggesting the use of the harmonic average, 
still with even weighting, which would indicate a fall by 20 per 
cent., would have been right, had he rested on the condition of 
the two classes being equally important at the second period 
only. But in this case also the arithmetic average with its 
proper weighting would make the same indication, and the 
geometric average with its proper weighting would diverge only 
by indicating a fall of 20.64 per cent. All this, however, is 
said only on the supposition that both these writers agreed in 
assuming that constant mass-quantities of each class were pur- 
chased at each period.* 

they agree), and the geometric average synthesizes these opposites (with weighting 
which likewise synthesizes their weightings, provided this composite weighting 
be even). Unfortunately this is exactly so only when the geometric average is 
restricted to being a mean proper. 

"^This assumption was used in Laspeyres's reasoning, as also that the classes 
were equally important at the first period, and so he happened to be right in his 
conclusion, confined to these conditions. But he never recognized these condi- 
tions, nor confined his argument to them, so that in general his position was no 
better than Jevons's. Had Jevons made the assumption that constant equal sums 
were spent on the two classes, his choice of the geometric mean would have been 
exactly right, as we have seen, — but not so his choice of the geometric average in 
wider cases. But he made this assumption only in connection with the harmonic 
average, which then is wrong. 



scRoi'K s mi-:th()I) 355 

On this iissumption we find Jcvoiis's prophecy fulfilled. He 
supposed another case, in which the price of one article remains 
unchanged at 1.00 and that of another rises from 1.00 to 2.00, 
and said that " the mean rise of price might be variously stated " 
as the arithmetic at 50 per cent., as the geometric at 41, or as the 
harmonic at 33, and added the sentence already quoted : " It is 
probable that each of these is right for its own purposes when 
these are more clearly understood in theory." ^ Strictly speak- 
ing, none of the averages has any purposes of its own, but each 
one is subject to certain conditions. Thus in the case supposed, 
with further assumption of the mass-quantities being constant, 
the arithmetic mean, indicatmg a rise of 50 per cent., is right if 
the two articles were equally important at the first period, the 
harmonic, with rise by 33 per cent., if they were equally impor- 
tant at the second period, and the geometric, with rise by 41 per 
cent., if they were equally important over both the periods. If 
their imjiortance was wholly vmeven, the mean rise might be any 
figure between and 100 per cent. — and the right figure would 
be indicated either by the arithmetic average with the weighting 
of the first period, or by the harmonic average with the weight- 
ing of the second period, or (approximately in many cases) by 
the geometric average with the weighting of both the periods.'' 

But we have anticipated somewhat, and must now seek to 
prove that this common result of the two averages always, and 
sometimes of all three, is the right one. 



III. 

§ 1. The reason why the argument from compensation by 
equal sums has been mistaken for an argument specially favoring 
the arithmetic average of price variations is because in making- 
use of it the advocates of this average have had in mind such 
conditions as are illustrated in these three schemata, — for the 
arithmetic price variations : 

5 B. 23, p. 121. 

® We have seen something similar in the preceding Chapter. But there the 
weightings corresponding to these were of another kind, and hidden (except in 
the third instance), so as not to be serviceable (being there, as here, inexact in 
the third instance). 



356 



THE METHOD FOR CONSTANT MASSES 



I 100 A @ 1.00 
II 100 A &, 1.50 



100 B @ 1.00 
100 B@ .50 



100 for [A] 
150 for [A] 



100 for [B] — 200, 
50 for [B] — 200; 



for the harmonic price variations 



I 100 A @ 1.00 
II 100 A @1.50 



100 B @ 1.00 
100 B@ .75 



100 for [A] 
150 for [A] 



100 for [B] 
75 for [B] 



200, 
225; 



for the geometric price variations 



I 100 A 
II 100 A 



1.00 
1.50 



100 B@ 1.00 
100 B@ .661 



100 for [A] 
150 for [A] 



100 for [B] — 200, 
66| for tB] — 216f 



in which, the classes always being supposed equally large at the 
first period, compensation by e(\uQ\ sums (and by equal prices) 
takes place only in the first example, where also the arithmetic 
average with even weighting indicates constancy. If, however, 
any one had wanted to use this argument for the harmonic aver- 
age of price variations, he might have made use of the following 
schemata, — for the arithmetic price variations : 



I 100 A @ 1.00 
II 100 A @ 1.50 



100 B^@ 3.00 
100 B^@ 1.50 



100 for [A] 
150 for [A] 



for the harmonic price variations 



I 100 A @ 1.00 
II 100 A (S, 1.50 



100 B^@ 2.00 
100 B' (a). 1.50 



100 for [A] 
150 for [A] 



for the geometric price variations : 



I 100 A @ 1.00 
II 100 A @ 1.50 



100 B^@ 2.25 
100 B'Cai 1.50 



100 for [A] 
150 for [A] 



300 for [B 
150 for [B 



200 for [B 
150 for [B 



225 for [B 
150 for [B 



— 400, 

— 300; 



— 300, 

— 300; 



■325, 
300; 



in which, the classes always being supposed equally large at the 
second period, compensation by equal sums (and by equal prices) 
takes place only in the second example, where constancy is in- 
dicated, even weighting being used, only by the harmonic aver- 
age. Again, if any one had wanted to use this argument for 
the geometric mean of price variations, he might have made use 
of these schemata, — for the arithmetic price variations : 



I 100 A @ 1.00 100 B^''@ 1.7320 
II 100A@1.50 100B^''@ .8660 


100 for [A] 
150 for [A] 


173.20 for [B] - 
86.60 for [B] - 


-273.20, 
-236.60; 


for the harmonic price variations : 






I 100 A @ 1.00 100 B^''@ 1.4142 
II 100 A @ 1.50 100 B^''@ 1.0606 


100 for [A] 
150 for [A] 


141.42 for [B]- 
106.06 for [B]- 


-241.42, 
-256.06; 



for the geometric price variations : 



8(;u()Im:'w mki'iioi) 



;io7 



[ 100 A @ 1.00 
II 100Ar'nl.50 



100 W'Oi} 1.50 
100 IV @ 1.00 



100 for [A] 
150 for [A] 



150 for [B] — 1>50, 
100 for [B] — 250; 



ill wliich, the classes always being supposed equally large over 
both the periods together, compensation by equal sums (and by 
equal prices) takes place only in the third example, where con- 
stancy is indicated, even weighting being used, only by the 
geometric average. 

The error in all these applications of the argument is mere 
ignoring of the fact that each of the averages with its own 
proper wcigliting applied to every one of these examples gives 
exactly the same results (except in those where the weighting 
for the geometric average is not even, its results then deviating 
somewhat). 

§ "2. But although we cannot use the argument to distinguish 
betAveen the three averages, we can use it to show the correct- 
ness of all three averages (the third, however, only in special 
cases), each with its own proper weighting — always on condition 
of constant mass-quantities. If we bring together the three 
simplest schemata on which the advocates of each average may 
rely, as follows — for the arithmetic price variations : 



I 100 A @ 1.00 
11 100 A (a), 1.50 



100 B@ 1.00 
100 B@ .50 



100 for [A] 
150 for [A] 



100 for [B] — 200, 
50 for [B] — 200; 



for the harmonic price variations : 



I 100 A @ 1.00 
II 100 A (ri\ 1.50 



100 B^ @ 2.00 
100 B^@, 1.50 



100 for [A] 200 for [B] — 300, 
150 for [A] 150 for [B] — 300; 



for the geometric price variations 



I 100 A @ 1.00 
II 100 A @ 1.50 



100B'^@ 1.50 
100 B'^(S\ 1.00 



100 for [A] 
150 for [A] 



150 for [B] — 250, 
100 for [B]— 250; 



we see the reason for the correctness of each average with its 
own proper weighting — always indicating constancy with perfect 
clearness only in the third example ; for in the others, although 
we have compensation by equal sums, it is on classes, and indi- 
viduals in them, that have different exchange-values over both 
the periods together. But, as already noticed, there is a cor- 
rection accompanying each deviation. In the first example, the 
class [B] is less important than the class [A] over both the 



358 THE METHOD FOR CONSTANT MASSES 

periods together ; but its price falls more than the price of [A] 
rises, since we know that a fall from 1.00 to .50 is greater than 
a rise from 1.00 to 1.50. And in the second case the class [B] 
is more important than the class [A] over both the periods to- 
gether ; but its price falls less than the j)rice of [A] rises, since 
we know that a fall from 1.00 to .75 is smaller than a rise from 
1.00 to 1.50. Still we do not yet perceive that in the former 
case the price of [B] falls exactly as much more as it ought to 
do, to make up for the smaller importance of its class ; nor that 
in the latter case, it falls exactly as much less as it ought to do, 
to allow for the greater importance of its class. But we may 
learn it with perfect certainty by the following reasoning. 

The first example may be converted into the following, al- 
ready used, by merely employing a different mass-unit of [B] , 
with prices inversely altered : 



I 100 A @ 1.00 57.73 B'''@ 1.7320 
II 100 A @ 1.50 57.73 B'' @ .8660 



100 for [A] 100 for [B] — 200, 
150 for [A] 50 for [B] — 200. 



Here the mass-unit of [B] is equivalent to the mass-unit of 
[A] over both the periods, so that A and B" may be taken as 
economic individuals. Now we purchase 157.73 such indi- 
viduals at each period, and we pay exactly the same sum for 
them at each period. Therefore our money has retained exactly 
the same purchasing power or exchange-value over, both the 
periods.^ And the second example may be converted into the 

1 The fact that 100 of these individuals are in [A] and 57.73 in [B] merely 
shows that [A] is —-^ = 1.732 times larger than [B]. This we already know 

from the fact that -v/^^;^;^ = 13 = 1.732053. With this weighting the geo- 
' lOU X 50 

metric average indicates a rise by 0.3 per cent., and is by so much wrong. — In 
Chapt. VIII. Sect. III. § 1 we saw inconsistency in the argument of the arithmetic 
averagist on the supposition of the classes always being equally important. But 
here the two classes, equally important at the commencement, are not thereafter 
equally important, and tlie inconsistency vanishes. Thus the arithmetic averagist 
argues that if [A] and [B] are equally important at the first period their com- 
pensatory variations should be to equal distances from their equal starting points. 
Suppose A rises from 1.00 to 1.50 and B falls from 1.00 to .50. Then [AJ is three 
times more important than [B], and therefore, starting from this position as a 
new first period, the price of § A should rise from 1.00 only one third as far as the 
price of 2 B falls from 1.00. This is precisely what takes place when A continues 
to rise from 1.50 to 1.51, while B continues to fall from .50 to .49. 



SCROPF/S MKTHOJ) 



:}.")9 



following^ also already used 



I 100 A @ 1.00 
II 100 A (<i)j 1.50 



141.42 V," @ 1.4142 
141.42 W («\ 1.0606 



100 for [A] 
ir>() for [A] 



200 for [B] — 300, 
150 for [B] — 800. 



Here also the mass-unit of [B] is e<juivalent to the mass-unit 
of [A] over both the periods, so that A and B'' may be taken 
as eeonomic individuals. Now we j)urchase 241.42 such indi- 
viduals at each period, and we pay exactly the same sum for 
them at each period. Therefore our money has retained ex- 
actly the same purchasing power or exchange-value over both 
the periods.^ 

§ 3. We thus obtain also here — that is, applicable only to 
cases with constant mass-quantities — a precise method of meas- 
uring the constancy or variation of the general exchange- value 
of money. It is : Find in all the classes, for use as units, 
masses that have the same money-value over both the periods to- 
gether, and measure the constancy or variation of the exchange- 
value of money inversely by the constancy or variation in the total 
sum of money needed at each period to purchase the constant num- 
bers of these ma^s-units supposed to be actually purchased. Nat- 
urally this method is not confined to cases with only two classes. 

In this method, however, the first part is superfluous, since, 
whatever be the mass-quantities in the various classes, we know 
that they must contain certain numbers of mass-units that are 
equivalent over both the periods, which numbers will be con- 
stant if the mass-quantities are constant ; but as we make no 
use of these numbers when ascertained, it is unnecessary to as- 
certain them. All we need, then, is to measure the constancy or 
variation of the exchange-value of money inversely by the con- 
stancy or variation in the total sum of money needed at each 
period to purchase the constant mass-quantities of all the classes. 

The formula carrying out this method, we may repeat, in its 
simplest form, is the following, 

^Here, [B] containing 141.42 and [A] 100 of these individuals, the former 
is 1.4142 times larger than the latter. This we also know from the fact that 

a'"^^-— ^ = ^ 2 = 1.414213. Witli this weighting the geometric average indicates 
a fall by 0.06 per cent., and is by so much wrong. 



360 THE METHOD FOR COJs'STANT MA8.SE.S 



P, _ a^ + b, + c, + 
P, a^ + b^ + Cj + 



or this, 

Pi a;a^ + 2//5i + 2ri + ' • ^ ^ 

(in which it is evident the sizes of the mass-units used have no 
influence). The last, to repeat also, we recognize to be the 
formula for Scrope's method, which, therefore, is the correct 
method for the cases in question. 

Now in the preceding Chapter (Sect. III. § 3) we found the 
method there discovered for cases with constant sums to be 
Scrope's method applied to the geometric means of the mass- 
quantities. But if we take the formula for that form of Scrope's 
method (there given as No. 8), and apply it to cases in which 
the two mass-quantities in every class, the one at the one period 
and the other at the other, are the same, the formula reduces 
to the last formula above. Or, reversely, by distinguishing x 
into x^ and x^, to represent the mass-quantities at each period 
notwithstanding that they are equal, and distinguishing the 
other symbols for the mass quantities in the same way, we may 
derive from the last expression this, 



Pj _ «X^r^2 + ^2^^3/12/2 + r2^^I^ 2 + 



P - .^ . . .^ ■ ;=--r— • (5) 



Thus Scrope's method applied to the geometric means of the mass- 
quantities is a comprehensive method, applying both to the eases 
with constant sums of money and to the cases loith constant mass- 
quantities. 

§ 4. Therefore, like the method examined in the precedmg 
Chapter for the cases with constant sums, this method for the 
cases with constant mass-quantities satisfies all the Propositions 
that more or less definitely prescribe what the variation of 
money in exchange-value in all other things, and consequently 
the inverse variation of the general level of prices, must be. 
Thus we see plainly that it indicates constancy if no prices vary, 
and if all prices vary alike it indicates the same variation (Propo- 



s("ropp:'s mkthod .'UH 

sitions XXVII., XLIV., XVII., XLV.), no matter what be 
the weights (provick'd, of course, they l)e such that, witli the 
]n'i(!e variations, they show the mass-quantities to be constant) ; 
nor can it indicate constancy if there is only one price variation, 
or if all price variations are in one direction — and in the latter 
event it of course cannot indicate a variation in the opposite 
direction (Propositions XX. and XXVIII.). It also satisfies 
Proposition XXXVI., if its own condition be observed in the 
omitted classes. This is so plain as not to need to be demon- 
strated. But again a similar remark has to be made here as 
was made in the corresponding passage in the preceding Chapter 
(Sect. III. § 4). That Proposition'' does not require the omitted 
class or classes to be of the same mass-quantities at both the 
periods. Yet this method requires that they should be ; for if 
they are not and its principal formula, (3), is extended to them 
(but altered, so as to distinguish between the new mass-quanti- 
ties at each period), the two results will not agree. Yet it is 
not this method which is being used and disproved, but another 
method ; and ^A/.v method satisfies the test oifered by that Propo- 
sition perfectly,^ 

§ 5. Moreover, as above seen, the arithmetic average of the 
price variations with the weighting of the first period and the 
harmonic average of them with the weighting of the second 
])criod are, under the given condition, universally the same as 
this method. Therefore these averages, with these weightings, 
are both correct. But they are superfluous, since Scrope's 
method is simpler and more convenient. 

And the geometric average with the weighting of both periods 
also reduces to this method in all cases when we are dealing with 
the variations of two equally (over both periods) important 
classes, so as to be able to use even weighting. This is so not 
only when the result indicated is constancy, but also when the 
result indicated is a variation. We have already noticed several 

3 Nor Proposition XXXII., when the prices are constant. 

* But again a similar qualification has to be added here also. What is said in 
the text may be said with reference to formulae either (1), (2), (3), or (4). But 
formula (5) satisfies the Proposition. In other words, this method satisfies the 
Proposition if we treat the additional different mass-quantities as this formula 
prescribes. 



J 



362 THE ^METHOD FOR CONSTANT MASSES 

instances of the former kind ; we may now notice a couple of 
the latter. These may be taken from the last set of schemata 
given above in § 1, where the labor has already been performed 
of obtaining not only classes, but individuals, equally important 
over both periods. In the first of these examples we see, there- 

fore, that we have p^ = ^^^ „ = 0.8660, indicating a fall of 

13.40 per cent. The geometric mean of the price variations is 

^ X o= ~o~ = 0.8660, indicating the same fall. In the 

second example we have p^ = ^. . ^ = 1.0606, indicating a 

J 3 3 3 

X — = — ^ 
2 4 \/2 

= 1.0606, indicating the same rise. 

In each of these examples it may be noticed that the vari- 
ation of general prices is from 1.00 to the same figure as in the 
price of B" at the second period. This relation is universal 
under the conditions supposed. These are that a/' =1.00, 
x" = y" and aja., = bjb.,. For from the latter, with the aid of 

a b., a., -i- b., a;"«/' 
other known relations, we derive ^ = — = — , - = .. '.. = 

\ \ \^\ y"N' 

y"l3^' x"a^' + y"^- P, Mq, a- ^- <^4-/V^ 

x"a^' ~ x"o.l' + y"^(' ~V~ M^^ ~ /9/' ~ o.(' ~ ^ ^ ~ «/' +/5/' 



So' fi 



\ i'l 

With uneven weighting the geometric average of the price 
variations does not necessarily agree with the common result 
given by the other two averages, each with its proper weighting. 
In showing that the other two averages always agree with the 
proper method we have shown that their common result is right. 
We need, therefore, not so much to show that the geometric 
average is wrong when it diverges, as to investigate its devi- 
ations. It is evident at once that the geometric average will 

^ Compare these with corresponding relations in the preceding Chapter, Sect. 
II. §4, Notes. 



DEVIATION OF THP: GEOMETRIC AVERAGE 363 

agree with the true method here in all cases corresponding; to 
the cases in which we have found that it would agree with the 
other true method, in the preceding Chapter (Sect. III. § 5). 
Some inferences similar to inferences in that chapter made about 
the deviations will also follow. 

IV. 

§ 1 . AVhat we desire to prove and to illustrate may be shown 
here on a single example. Let us suppose the class [B] to be 
twice as imj)ortant, over both the periods together, as the class 
[A]. The price of A rising from 1.00 to 1.50, to have con- 
stancy according to the geometric average with this weighting, 
the price of B must fall from 1,00 to v^f = .8165 ; and in order 
to have this weighting with these price variations, the mass- 
quantities must be in the following proportions : 

I100A@1.00 271.08B@1.00 I 100 for [A] 271.08 for [B] — 371.08, 
II 100 A® 1.50 271.08 B@ .8165 1 150 for [A] 221.34 for [B] —371.34, 



in which v/100 x 150 = | v/271.08 x 221.34. Here Scrope's 

371 34 
method shows a rise of prices, viz., ^ = 1.000698 — a 

rise by 0.0698 per cent. That this is right is more apparent 
upon rearranging the mass-unit of [B] as follows : 



I 100 A @ 1.00 200 B'' @ 1.3554 
II 100 A @ 1.50 200 B'' @ 1.1067 



100 for [A] 271.08 for [B] — 371.08, 
100 for [A] 221.34 for [B] —371.34. 



For here the mass-unit of [B] is of the same exchange-value 

over both the periods as the mass-unit of [A] ; wherefore, as we 

have to pay more for these equivalent mass-units at the second 

period, it is evident that their prices have risen. Thus Scrope's 

method is right in its indication, and the geometric method errs 

by indicating constancy when it ought to indicate a slight rise, 

and so, in this case, it is slightly below the truth (by 0.06976 

per cent.). 

In this example the figures for [B] could also be arranged as 

follows : 

I 180.72 B' @ 1.50, 
II 180.72 Bi@ 1.2247, 
or 



364 THE METHOD FOR CONSTANT MASSES 

I 221.34 Bii@ 1.2247, 
II 221.34 B" @ 1.00, 

these falls being the same as from 1.00 to .8165, and the sums 
spent on [B] being the same as before. Or we could take half 
of each of these, as follows : 

I 90.36 Bi @ 1.50 + 110.67 B" @ 1.2247, 
II 90.36 Bi @ 1.2247 + 110.67 B" @ 1.00 ; 

and here the price of [B] has fallen from 1.50 to 1.00, in- 
versely as the price of A has risen from 1.00 to 1.50. But it 
is not equal masses of [B] that hand on this fall. We virtu- 
ally have three classes. And in the quantities indicated the 
three classes are equally important over both the periods ; and 
the geometric average and the true method give the same diver- 
gent results as before. 

Now suppose another distinct case, in which it happens that 
at both periods we purchase 100 A, 100 B, and 100 C, and sup- 
pose the price of A rises from 1.00 to 1.50, and the price of B 
falls from 1.50 to 1.2247 and the price of C from 1.2247 to 
1.00. Here the classes [B] and [C] together make a fall ex- 
actly the reverse of the rise of the single class [A] , and as the 
individuals in each of these classes are equal in number, every 
rise of 1 A from 1.00 to 1.50 seems to be met by a fall of 1 B 
and 1 C from 1.50 to 1.00. Therefore we should expect con- 
stancy. And constancy is indicated by Scrope's method, which 
shows that 

P, 150 + 122.47 -H 100 
p^ - 100 + 150 + 122.47 ~ 

But if we apply the geometric method to these conditions we 
must give these weights to the classes — to [A] n/1.00 x 1.50 
= 1.2247, to [B] v/1.50 x 1.2247 = 1.3554, to [C] 
n/1.2247 X 1.00 = 1.1067, and now 

2 _ '•'|/^(|)1.2247 ^ (^ ^2 y. 3554 +1.1067 _ 0.999302 , 

indicating a fall by a trifle less than 0.069 per cent. To get 
constancy here by the geometric method we should have to 



DEVIATION OF THE GEOMETRIC AVERAGE ,'365 

weight the common fall of [B] and [C] as 2 to the rise of [A] 
as 1. This wc coiikl do if it were proper to weight [B] and 
[C] by donbling the geometric mean of their weights, the class 
[B] being as mnch more imj)ortant as tlie class [C] is less im- 
portant than the class [A]. But we can form no general prin- 
ciple of weighting of this sort. For instance, in the preceding 
example this kind of weighting would still weight the class [B] 
as two to [A] as 1 , and yet with this weighting the geometric 
average there showed about the same error. 

Hence it appears that in these complex cases, even on the 
principle of the geometric method itself, Scrope's method is cor- 
rect, and the geometric average of price variations, with the 
best weighting we know of, is wrong so far as it diverges. 

§ 2. In the examples reviewed the preponderating class or 
classes have been the ones that fall in price, and the geometric 
average has been found to err below the truth. Making [A] 
the larger class, we should find the error to lie on the other side. 
These facts, added to ^vhat we already know about the deviation 
of the geometric average,' lead to the inference also here that 
when the prices that rise above the general average are those of 
preponderating classes, the geometric average of price variations 
yields a result beloio tJie truth ; and when the prices that fall below 
the general average are those of preponderating classes, it yields a 
result above the truth. 

Luckily, as we have another method which is not only ex- 
actly correct but far more convenient, we are not so much 
interested in the error of the geometric method as we were in 
the preceding Chapter. We m.ay be sure, however, here as 
well as there, that w^ ith moderate price variations such as usually 
take place, the geometric average will not much deviate from 
the truth. 

§ 3. Some extraordinary cases deserve a moment's attention. 
Suppose the classes [A] and [B] are equally important at the 
first period, and the price of A rises from 1.00 to 1.99, to what 
figure ought the price of B to fall from 1.00 in order to com- 
pensate for that rise ? — always supposing that the mass quanti- 

1 Cf. other instances in Notes 1 and 2 in the preceding Section. 



366 THE METHOD FOR CONSTANT MASSES 

ties are constant. We can now answer without hesitation : it 
ought to fall to .01 ; for the arithmetic average with the even 
weighting of the first period indicates constancy under these 
price variations, and this average with this weighting is correct.^ 
Now we might expect, from the reasoning in Chapters VII. and 
VIII., that according to the geometric method this fall would be 
made out to be too great. On the contrary, the conditions are 
not what were there supposed, and they require the geometric 
average, applied to these conditions, to indicate a rise — and, be- 
cause of the enormous variation m the [)rice of B, with conse- 
quent influence upon the relative sizes of the classes, a consider- 
able rise — in the general level of prices, meaning that there has 
not been sufficient compensation. In fact we find that the geo- 
metric average here indicates a general rise by 40.16 per cent. 
And if we suppose that B falls from 1.00 to .01, and want the 
compensatory rise for A so as to make money constant in ex- 
change-value, instead of requiring A to rise from 1.00 to 1.99, 
the geometric method requires it to rise from 1.00 only to 1.463. 
The truth is that, when dealing with such enormous price vari- 
ations of the smaller class, as shown in similar instances in the 
preceding Chapter, the geometric method becomes wholly un- 
workable. To show this we may compare the results given by 
the geometric average with its proper weighting with the true 
results, indicated by the arithmetic average with even weighting, 
when, the classes [A] and [B] being equally large at the first 
period, and the mass-quantities remaining constant, the price of 
A is supposed in all cases to rise from 1.00 to 1.99, and the 

2 As is rendered plain by the following schema : 

I 100 A @ 1.00 7.092 B" @ 14.10 I 100 for [A] 100 for [B] — 200, 
II 100 A @ 1.99 7.092 B" @ .141 | 199 for [A] 1 for [B] —200, 
in which the mass-units, A and B", are equivalent over both the periods together, 
and the same total sum is paid at each period for purchasing these equivalent 
mass-units. (To have even weighting over both the periods together, the condi- 
tions would have to be : 

I 100 A @ 1.00 100 B @ 14.10 I 100 for [A] 1410 for [B] — 1510, 
II 100 A @ 1.99 100 B @ .141 | 199 for [A] 14.10 for [B] — 213.10. 

Here the geometric average with even weighting, 1^1.99 X .01 = 0.1411, indicates 

a fall of 85.89 per cent.; but also the arithmetic average with the weighting of 

213.1 
the first period indicates the same fall, for -^-^ = 0.1411.) 



DEVIATION OF THE OKOMETRIC AVERAGE 307 

price of B is su|)[)()SO(l to fall variously to the following- tigures, 
stated ill the first column, the true average being stated in the 
next column, aud the geometric average in the last : 

1.00 1.495 1.4958 

.81 1.400 1.4021 

.64 1.315 1.3200 

.49 1.240 1.2502 

.36 1.175 1.1947 

.25 1.120 1.15()3 

.16 • 1.075 1.1403 

.12 1.055 1.1436 

.09 1.040 1.1562 

.04 1.015 1.2250 

.01 1.000 1.4016 

.005 0.9975 1.4954 

Here the geometric method remains approximately correct till 
^.^ descends beyond .50 and its variation becomes the greater 
of the two, while [B] becomes less than half as large as [A] 
over both the periods, after which it departs appreciably from 
the truth. A strange thing is that after y^^' reaches about .15, 
the further it descends, the more the geometric average rises. 

An objection previously urged against the arithmetic average 
was that with even weighting it permits of no compensatory fall 
of B after A has risen to 2.00. That was on the supposition of 
even weighting over both the periods together. With condi- 
tions permitting of even weighting over both the periods to- 
gether there should be possible a compensatory fall of B for 
every rise of A. But if the weighting is even only at the first 
period, and if the sizes of the classes rise and fall with the rise 
and fall of their prices, there being no change in our purchases 
of them, there is no reason why beyond a certain point in the rise 
of A there should be no compensatory fall of B possible. That 
objection against the geometric average was accompanied by an 
argument for the geometric average. We now see that under 
the circumstances supposed the geometric average is in this re- 
spect no better than the arithmetic, each with its proper weight- 
ing. It is even worse, as it takes aAvay the possibility of com- 
pensation at an earlier point in the rise of A. Moreover it is 
peculiar in that after A has ]iassed the point where the possi- 



368 THE METHOD FOE CONSTANT MASSES 

bility of complete compensation ends, the descent of B gains its 
maximum compensation before reaching zero. 

Y. 

§ 1. In closing this Chapter some tests may be employed 
similar to those at the end of the preceding. 



I 100 A @ 1.00 100 B @ 1.00 

II 100 A @ 1.50 100 B@ .60 

III 100 A @ 1.00 100 B @ 1.00 



100 for [A] 100 for [B] — 200, 
150 for [A] 60 for [B] — 210, 
100 for [A] 100 fgr [B] — 200. 



Here it is evident that the exchange- value of money is the same 
at the third period as at the first, whatever it be at the inter- 
mediate stage ; and this constancy at the third period compared 
with the first should be indicated by the results of the two vari- 
ations. The weighting at the first and at the third ' period is 
even, but at the second period it is 5 for [A] and 2 for [B] . 
Now if we use the arithmetic average in both measurements, 
and each time with the weighting of the earlier or first period 
of the two compared, that is, in the first price variations with the 
even weighting of the first period, we get an indication of a rise 
of 5 per cent. (i{| + |} = |i = 1.05) ; and in the second 
price variations with the uneven weighting of the second period 
we get an indication of the inverse fall of 4.77 per cent. 
(1 {5 X I -h 2 X f I = If = 0.9523) ; and these two together 
indicate the same level of prices at the third period as at the 
first. And exactly the same indications are given by the har- 
monic average of the first price variations with the weighting of 

/ 7 21\ 

the second period ( ^ ^ ^ = -- J , and by the harmonic 

average of the second price variations with the even weighting of 

/ 2 20\ 

the third period ( j — g = ^ I — as also directly by Scrope's 

method of comparing the total sums.^ And, once more, the correct 

1 The same results could also be obtained by combining these averages diflfer- 
ently and using the same weighting throughout : — thus either by using even 
weighting (of the earlier and of the later period) with the arithmetic average of 
the first variations and the harmonic average of the second ; or by using the 
weighting 5 for [A] and 2 for [B] (that of the later and of the earlier period) 
with the harmonic average of the first variations and the arithmetic average of 
the second. But these are methods above rejected. 



TMST CASKS :U)U 

lliiiil result is obtained also l)y using the geometric average, in 
both cases with the same weighting, which is l.oSll for [A] 
and 1 for [B] . With this weighting the geometric average of 
the tirst price variations is "'[/ (1)*"^*" X |= l.OoKJ, indicat- 
ing a rise of o.KJ per cent., and of the second, "•■''|^^(2y.'^Mi ^^ 6 
= O.iloOS, indicating a fall of 4.92 j)er cent. It would seem as 
if the same weighting ought to be used in both the averagings, 
since in each set of variations the only difference in the weight- 
ings is tlie order of their occurrence. Still the geonu^tric aver- 
age does not give such true indicatiou in each set of price vari- 
ations as do the other two methods.^ 

We perceive, however, that the error of the geometric method 
above the truth in the first price variations, where the rising 
l)rice is of the predominating class, is exactly counterbalanced 
by the error of the geometric method below the truth in the 
second price variations, where the variations are revcn'sed and 
the falling price is of the predominating class ; for 1 .051 (J : 1 .05 
:: 0.9523 : 0.9508. 

If the weights were uneven at the first and third periods, oi- 
altogether so different as to make the classes very unequal in 
size over both the periods together, in each comparison, while 
the price of the smaller class varies considerably, the error of 
the geometric method al)ove or below the truth in each measure- 
ment would l)e greater than in the case here cited, and might be 
considerable. Yet the same counterbalancing Avould always 
take place, and the indication for the third period through the 
intermediate one would still indicate constancy. 

§ 2. With two or more intermediate periods the geometric 
average does not necessarily give the right indication for the 
last period. For example : 

I 100 A @ 1.00 100 B @ 1.00 



II 100 A @ 1.50 100 B@ .75 

III 100 A @ .06* 100 B @ 2.00 

IV 100 A @ 1.00 100 B @ 1.00 



■100 for [A] 100 for [B]— 200, 

150 for [A] 75 for [B]— 225, 

66f for [A] 200 for [B]— 266ff, 

100 for [A] 100 for [B]— 200. 



- The true final result is also obtained through intermediate errors in two other 
ways — namely, by averaging both sets of variations aritlimetically with the 
weightings of tlie later periods, or harmonically with the weightings of the 
earlier periods. But these are methods also above rejected. Thus there are seven, 
different ways of obtaining the known final result. 
24 



370 THE METHOD FOE CONSTANTi'SUAIk) 



Here the sreometric method gives the following indications : 

1 St variation '-'p' (fp^^ x | =1.1256, 
2d variation -p'' r^lff^'^' = 1.1917, 
3d variation "j//|^7J)i--^ = 0.7475. 

These form the series of index-numbers 100, 112.56, 134.14, 
100.27 ; and so this method is wrong at the end by an error 
0.27 per cent, above the trnth. As indicated in the last column, 
Scrope's method gives the index-numbers 100, 112.50, 133.33, 
100; which show the correct variations to be: 1st 1.125, 2d 
1.1852, 3d 0.75. 

In Scrope's method is perceived the merit, as there has al- 
ready been occasion to remark, that it universally stands Pro- 
fessor Westergaard's general test.'^ It is always consistent with 
itself in direct and in indirect comparisons over any number of 
periods. It stands one and all of the tests to which we can sub- 
ject it. Confined to the cases when the mass-quantities are con- 
stant, this method is absolutely perfect. And, consequently, so 
also the two methods of averaging the ])rice \'ariations into which 
it can be exactly analyzed. 

Therefore, possessed of this perfect yet simple and most con- 
venient method, we liave no interest here in pursuing further 
the investigation of the slightl}' deviating error in the geometric 
method. We may, however, — to extend a remark already made 
— be suiAe, here as well as in the preceding Chapter, on account 
of the many counterbalancing influences, the geometric method 
is not likely in a long series to be far wrong at any period. 

St'e Chapt. V. Sect. V[. iJ 7. 



CHAPTER XII. 
THE UNIVERSAL METHOD. 



§ 1 . The conditions presupposed hv both the precediujj urgu- 
nieuts are unlikely to be met with in practice. The first argu- 
ment, in supposing the sums of money to be constant in sjiite of 
th(! j)rice variations, can have application only if prices vary 
through changes in supply. The second, in supposing the mass- 
(juantities to be constant in s])ite of the price variations, can 
liave application only if prices vary through changes in de- 
mand. Neither of these changes is likely to take place alone. 
Prices generally vary through changes both in su})]>ly and in 
demand. Both these conditions must be admitted ; and only that 
argument is complete which takes them both hito consi(U'ration. 

Or the user of the first argument can defend his position only 
by claiming that the weighting should be according to the smaller 
money-values (or, more properly, the smaller exchange- values) 
at either period. And the user of the second argument can de- 
fend his position only by claiming that the weighting should de- 
pend upon the smaller mass-quantities at either period. Each 
nnist rely upon something that is common to both the periods, 
eliminating all the rest of the same kind. 

The practical and theoretical objections to these positions ha\'e 
ah'eady been set forth in Cliapter IV. Section V. Their im- 
j)ropriety is especially apparent when we place them side b\- 
side. For which of them has more reason for it than the other? 
If we want the same material world at both periods, do we not 
equally much want the same economic world at both periods? 
We can not have both of these things together. And trial 
shows that (m posited variations of prices and of mass-(piantities 

371 



372 THE UNIVERSAL METHOD 

the two methods give very divergent results — sometimes the 
former giving the higher result, and sometimes the latter, and 
more or less so, and sometimes both giving nearly the same re- 
sults, without order or principle. And to use both, drawing 
some mean bet^veen their results, would convey no special mean- 
ing, and has no justification. 

Or again, the user of the first argument might defend his posi- 
tion by drawing a mean between the total money-values of a 
class at each period, and treating this mean total money- value 
as if it were the total money-value at each period, — and doing 
so with every class, he would weight them accordingly. And 
the user of the second argument might defend his position like- 
wise by drawing a mean between the total mass-quantities of a 
class at each period, and treating this mean total mass-quantity as 
if it were the mass-quantity at each period, — and doing so with 
every class, he would employ these mass-quantities as the basis 
of his weighting (or would simply apply to them Scrope's 
method). 

Now if the user of the first argument employed the geometric 
mean between the total money -values of every class at each of 
two periods as the weight of every class, he would be doing 
merely what lias already been recommended in Chapter IV. 
He would do well, however, to avoid recommending his pro- 
cedure on the ground of an as if. The total money- values of 
the classes are different at each period, and what we ^^-ant is not 
what might have been a similar state of things if a mean total 
money-value had existed in every class at both periods ; but, as 
explained in the earlier Chapter, what we want is the number 
of individuals in every class that, given the facts as they are, 
have the same exchange-value over both the periods. And if 
the user of the second argument employed a mean between the 
total mass-quantities at each period, he would be employing a 
method also admitted in that Chapter as a tenable one. But be, 
too, ought to find some other reason for his position than an as 
if. As for the kind of mean to use here, evervthmg we have 
learned points to the geometric mean as the proper one. But 
we shall later find that the arithmetic mean gives almost similar 



TWO METHODS SUG(3^ESTED 'MS 

results, wluircfore its *>reatcr convenience is a reconimondation 
for its practical employment. 

§ 2. Thus all tlie methods as yet in these pages examined that 
have any claim to consideration as theoretically reasonable and 
as likely to yield truthful, or nearly true, answers, reduce to 
these two : (1) the geometric averaging of the price variations 
with weiuhtinii" accordinj'' to the ti'cometric mean of the full total 
money-values at both periods — which, as before, for brevity, we 
shall call simply the geometric method ; and (2) Scrope's method 
applied to the geometric means of the full mass-quantities at both 
periods — whi(;h, again for brevity, we shall call Scroj)e's emended 
method. 

For the first of these the recommendation is that it takes into 
consideration the conditions at both periods, correctly weights 
the classes and uses the best average we have for averaging the 
price variations of the economic individuals, whose relative num- 
l)ers have been determined in the weighting. The objection to 
it is that the geometric average ceases to possess the virtue of 
the geometric mean when it is applied to more than two classes, 
or to two une(pial classes. It is true that all commodity-classes 
fall into two general classes : those whose prices rise above the 
average variation, and those whose prices fall below it, — not to 
mention a third class, whose prices vary with the average, the 
presence of. which class is inditferent. And in i)ractice those 
two general classes may ni<jstly l^e nearly equal in size. Hence 
in practice the geometric average is not likely to depart much 
from the truth. Still, we have seen that when the classes are 
very unefpial and tlie price variations are very great, this aver- 
age may deflect considerably. Therefore we should prefer not 
to have to rely on this method alone. 

The second of these methods has in its favor that it likewise 
takes into consideration the conditions at both periods, and that 
it avoids the use of the geometric average, using only the geo- 
metric mean, so that it escapes perversion arising from that 
source in the extraordinary cases of great inequality in the sizes 
and of great variation in the prices. The principal argument 
for it is that it lias been found to be the fundamental method 



374 THE UNIVERSAL METHOD 

underlying both the partial methods. For, being the right 
juethod both for constant money-sums, when the mass-quanti- 
ties vary, and for constant mass-quantities, when the money- 
sums vary, why should it not be extendible to all cases, and be 
the true universal method? Unfortunately only one of those 
partial methods was found to be perfect. And the very one by 
which this method is suggested was found not to be perfect. 
Hence there is a probability of this method also not being perfect 
in all cases. 

There is, however, still another way of deducing a universal 
method from the two partial methods, — adding a third method 
with claim upon our attention as likely to give approximation 
to the truth. This new method, by slightly altering each of 
those methods in the same manner, applies them to all possible 
cases, getting the same result always, whether it modifies the 
one or the other. And this alteration, or modification, in each 
case, respects the principle of simple mensuration, that we must 
at both periods be dealing with the same whole, or with similar 
wholes, allowing the details to be diiferent. Hence this method 
seems to have much in its favor. Since it modifies the imper- 
fect one, there is some hope that it escapes the imperfection. 
Whether it does so, will be seen in the sequel. We must now 
develop this method. Then we may test it, and compare it 
with the other methods. 

11. 

§ 1. Both the partial methods examined in the preceding 
Chapters start out with the same injunction, that we should ^».f? 
hi all the classes mass-units that have the same exehange-valae 
over both the periods together. This having been done, the first 
method, applied to conditions where the sums of money spent 
on the classes are not the same at both periods, becomes this : 
measure the constancy or variation of the purchasing poiver of 
money by the constancy or variation in the total number of these 
mass-units purchased or purchasable at each period loith a given 
total sum of money spent in the same proportions as the total sums 
actually were spent at each period. Of course we may take one 



A 'I'llll.Mi MF/I IIOI> 'i75 

of the total sums as it actually was, and reduce the other to it. 
The second method, applied to conditions where tlie mass- 
(juantities purchased of the classes are not the same at both 
periods, becomes this : mafsmr fJic i-oiisfaiici/ or rariafion of flic 
av('Ji(mf/e-v((lne of tii(>n('(/ invrrsclt/ hi/ the ('o)isf(nici/ or niriation 
in the totiil .sum of money needed (d each period to purchase a (jiven 
total numher of these mass-unUs in the same proportions an tin- 
total numbers of them actuafli/ were purchased at each period. Of" 
course, again, avc may take the one of the total numbers as it 
actually was, and reduce the other to it. The measurement of 
the constancy or variation of prices in li^eneral is tlie inverse of 
these, that is, by inversion in the first case and without inver- 
sion in the second. Both these methods, to repeat, applied to 
the same conditions where both the sums of money and the mass- 
quantities vary from [)eriod to })eriod, yield the same result. 

These two methods, which really form one bipartite method, 
call for illustration first by numerical examples. Suppose we 
find this state of things : 

r 90 A @ 1.00 70 B (<r 1.00 j 90 for [A] 70 for [B], 
II SO A @ 1.50 150 R 03 .50 | 120 for [A] 75 for [B], 

This admits of conversion into the followinir : 



T90Ar'n 1.00 40.415 B'-'f?*) 1.732 — 180.415 
II 80 X(o\ 1 .50 8().n05 V/'(<t}. .866 — 166.605 



90 for [A] 70 for [B] — 160, 
120 for [A] 75 for [B] — 195, 



in which the mass-units, A and B", have the same money- value, 
and the same exchange-value, over both the periods together. 
Now at the first period we bought 1 30.41 5 such mass-units for 1 60 
money-units, and at the sec(»nd we bought !()(). (JOo such mass- 
units for 195 money-units. Therefore at the second period we 
could buy, spending our money in the same proportions as we 
did then actually spend it, 136. 701 such mass-units for 160 
money-units (as we learn by the simple use of the rule-of-three). 
Thus the purchasing power of 160 money-units has risen from 
purchasing at the first period 1.30.41") to purchasing at the 
.second ])eriod 136.701 mass-units that are equivalent over both 

the periods. Hence its variation lias been ' =1.0482; 
^ 130.415 ' 



;>7() THE UNIVERSAL METHOD 

and the variation of prices has been ,^— "^TTr-r = 0.9540, indi- 
^ 136.701 

eating a fall of prices by 4.60 per cent. And, again, at the 
iirst period we gave 160 money-units for 130.415 such mass- 
units, and at the second we gave 195 money-units for 166.605 
f^uch mass-units. Therefore at the second period we gave 152.64 
money-units for 130.415 such mass-units, in the same propor- 
tions as we actually did then purchase them. Thus this mass- 
quantity of 130.415 mass-units equivalent over both periods has 
fallen in price from 160 at the first period to 152.64 at the 

152.64 
second, and the variation of the general price-level is 

= 0.9540, indicating a fall by 4.60 per cent., the same as be- 
fore. It may be added that on these data the geometric method 
indicates a fall of prices by 4.49 per cent.;^ and Scrope's 
emended method indicates a fall by 4.71 per cent.^ 
Another example may be supposed as follows : 



75 for [A] 70.71 for [B], 
90 for [A] 100 for [B]. 



I 75 A @ 1.00 70.71 B@ 1.00 
11 60 A @ 1.50 133i B@ .75 

This may be converted into 

I75A@1.00 50 B'^@ 1.4142—125 | 75 for [A] 70.71 for [B]— 145.71, 
II 60 A@ 1.50 94.28B'''@ 1.0606—154.28 90 for [A] 100 for [B]— 190.00, 

in which the mass-units, A and B", are equivalent over both 
the periods. At the first period we got 125 such mass-units for 
145.71 money-units, and at the second 154.28 of them for 
190.00. Therefore, in the same proportions, at the second 
period we should get 118.31 such mass-units for 145.71 money- 
units ; and so the purchasing power of this sum of money has 
fallen from purchasing 125 to purchasing 118.31 ; and inversely 

125 
the general level of prices has varied thus : ,. .. „ ^-. = 1.0565, 

which indicates a rise by 5.65 per cent. Again, at the first 
period we paid 145.71 money-units for 125 such mass-units, 
and at the second 190.00 for 154.28. Therefore at the second 

1 This is slightly above the other result. The weights are 1.4342 for [A] and 
1 for [B] . Notice that the rising price is of the preponderating class. 

2 On the arithmetic means Scrope's method indicates a fall by 6.41 per cent. 



i A Til 1 1,' I) ^r^;Tl!()l) .377 

period we had to pay, in the same j)r(>porti(iiis in whicli we 
actually did sj)eiid our money, 1 ").'i.94 money-units for 125 mass- 
units. Thus the price of 1 2.5 such mass-units, which are equiv- 
alent over both the periods, has risen from 145.71 to 153.87, 

... 153.94 ,., . . ,. . 

and tlie price variation is ,^^ =1.();)()0, likewise indicating 

a rise hy 5.()5 per cent. Here, we may again add, the geometric 
method indicates a rise by 5.64 per cent.; '^ and Scrope's method 
also indicates a rise by 5.(34 per cent.^ 

§ 2. That tliese two ways of making the calculation univer- 
sally agree, and coalesce into one method, may be demonstrated as 
follows. Employing our usual symbols, we construct this com- 
prehensive schema, representative of any possible state of things : 

I 'i A ® a, .i/iB@/3, ZiC@)', — ■'i + 2/i + }i + 

1 1 ^:j A (I'D a, y, B @ / 1, z, C f<\ }'., — x^_ +!/■, + )■, + 

rio, for [A] yj\ for [B] 2j}-, for [<'] — 2\a^ 4 yA + ^O'l + 

1 ^.^a^forCA] y,A for [B] 2,}., for [C] — -v.,"-. 4" ^23-2 + V/2 + , 

in which the mass-units, A, B, C, , of the classes, [A], 

[B], [C], are the customary commercial ones, and x, y, 

2, , represent the numbers of them that are bought and sold 

at the periods indicated by the numerals attached to them, and 

a, /9, ;-, , represent the prices of these mass-units at the 

periods similarly indicated. The number of classes may be ex- 
tended indefinitely, but must be the same at both periods. As 
the mass-units are various, we must reduce them to equivalence 
over both the periods. We may take the mass-unit of any one 
of them, say of [A] , as our unit, and reduce the rest to equiva- 
lence with it. Represent the mass-unit of [B] that is equiva- 
lent over both periods to A by B," and its prices at each of the 
periods by [i^" and ^i," respectively ; and treat the equivalent 
mass-unit of [C] in the same way. The first half of the above 
schema then becomes the following : 

^ Alnio.st exactly like the other, but, this time, slightly below. Notice that 
the weighting is almost even, but slightly preponderating on the side of the fall- 
ing priee, the weight for [A] being 82.1tJ and that for [B] 84.09. 

* These, however, differ in further decimals. For their closeness the near 
evenness of the weighting will also be found to be the reason. — On the arithmetic 
means Scrope's method imlicates a rise by 4.S() per cent. 



378 THE UNIVERSAL METHOD 



I X, A (S) a, ^ B- (^ H- p C- (m y- , 



5, 



II X, A @ «, IV' B'" @ ^i' !^' ^" @ '/' 



Here the conditions are that l-i^'^-i.^' = yl'X-l' = = 'Wh y ^"i<i 

^ // ^ ., // 
also that 'yj, = y , -77 = - , and so on. From the first coudi- 

3 " a a 
tion we derive '—j, = ^^ ; wherefore, by means of the second 

condition, ^ = yyyl, whence /3/'^= -^^^- I^^ '^ similar man- 

ner we obtain i3^'-'= "™ . Thns we have /?/'= J^^i , and 

/?/'= J^^; wherefore 



and 



And treating the prices of [C] in the same way, we obtain 



and 






^2 

/ 2 



and so on with all the other classes. By substituting these 
values in the last schema, it becomes : 



a^a^ ' OtC, 



II X, A® a, y, V'^B^^ @ (i\ .,V^_ C^^ @ y/^ 






ffjCj 



A Tinui) MKTiroT) '.>m 

the .second half of the schema remaining the same as in the first. 
Here A, B", C", are mass-units e(i[aivaleiit over both the 

periods. Now at the first period we get -'"i + .'/i ^ ' ^ + 
z^^ ' " + such mass-units for A\a^-\-l/^j3^-\-ZJ^-^- raoney- 

12 

22^''' + ^iich mass-units for' x^a.^-^i/.j'^,-^-\-z.j.^-^ money- 
units. Therefore, in the same proportions, we should get at the 
second period (as we learn by the rule-of-three) 

(•>-.+ y . J'!:^ + n Jg;; + ) (^i«.+ yA+ V. + ) 

■'^'2«2+ yA+ \r2+ 

sucli mass-units fora'^aj -|- y^[i^ -\-'^\T]. + money-units. Thus 

the purchasing power of this sum of money has varied from 
purchasing the former number to purchasing the latter number 
of mass-units equivalent over both the periods ; and the pur- 
chasing power of all sums of money, or of money in general, 
that is, the exchange-value of money in all the other things, has 
varied in the same proportion. And the general level of prices, 
varying inversely, has varied as is thus represented, 

""' (-' + >-A/!'r";+ Wig + ) (».'■ + y.ft + ^,h + ) 

(^ + y^^'$+ Wi^: + ) (-.". + y,^ + '>y< + ) 

Again, at the first period wegive.rj«j -f- //,/9j +z^y^ + money- 



mass- 



380 thp: uniyersai. method 

units for x, + i/, ^j - " + z, ^ — " + such mass-units, and 

at tlie second period we give x^a^ + y.jS^ + ^-iT-z + money- 

'-'-^ -f -^o ^ M-^ 4- such mass-units. 

Therefore, in the same proportions, we should at the second period 
have to give 

(-^^+^^A/^+wg:+ ) (^.+1/^+^.7.+ ) 

,^+,.,^M^^.Jm. + 

money-units for x^a^ + ij^ J^ + z^ ^^^ -f- such 

units. Thus the price of this number of mass-units equivalent 
over both periods has varied from the first to the second sum, 
and in the same proportion has varied the price of any given 
number of such mass-units purchased at each period in the same 
proportions as each of the totals at those periods actually were 
purchased. Therefore 

p. ^ {'^ +"■ V'^: + " Vg + ) {v'+y^^- + '■>■' + ) 

""' (■"+!"a/«-:+W„^:+ ) (.v,+y.3,+v,+ )' 

which is the same as the preceding, Q. E. D, 

Above, in Chapter IX. (Section 1.) we criticized the argu- 
ment from compensation by equal mass-quantities and the argu- 
ment from compensation by equal sums of money because, 
although modelled on correct methods of measuring variations in 
a particular exchange- value of money, which give the same result 
from two opposite points of view, those arguments, as hitherto 
employed, gave diiferent results, disclosing error somewhere. 
We then saw that the error lay in the method usually adopted 
of employing the argument from compensation by equal mass- 
quantities. Thereupon we determined the right method of em- 
ploying the argument from compensation by equal mass-quan- 



A Tllllil) MKTIIOI) ;J81 

tities, Wc now Hud tlmt this rit>lit use of this ariiiinu'nt and the 
right use of the other urgiiinent yield icU'utieal results. We 
have, therefore, apparently, got rid of the error whieh made 
those two arguments seem opposed to eaeh other, 

§ 3. We have done more still. We have finally reaehed a 
universal formula such as we have been seeking. For the above 

P., 

expression for ', twice obtained, is a universal fornuda for the 

constancy or variation of the general level of j)rices, and the in- 
verse of it is a universal formula for the constancy or variation 
of the general exchange- value of money in all other things. 

The above expression may be sim|)litied, and, being re- 
arranged so as to be brought into coufonuitv with certain other 
formuhe, it may be written thus, 

p, ■'•,«, + .'/i/^i + ■'■y'^''-: + //..^w^, + ' 

And this ma^■ be restated thus. 



p., a, + b, -h 



P,_a, + b^ ,.^, ,,., 

P. a, + b.-h \a^ 1:9; ' ^-^ 

or thus, 

Ja,a..^;+Jb,K-^; + 

s|a,a,-- + ^b,b,- + 

Both these, like the corresponding formuhe in Chapter X., 
Section III, § 3, namely those numbered (6) and (7), are more 
curious than useful. Here we have no additional form corre- 
spcHiding to formula (8) in that place. The nearest we can get 
to that formula is by altering the last into this, 

P, _ a, + b, -h ''- V'"'-^'^^ -^'^W^'^^S^ -^ 



P, a, -f- b, + 



«i>J-^V^'2a'"^^'\-^'-'^^'b'"^ 



382 THE UNIVERSAL METHOD 

Thus this method is distinct from Scrope's method in any of its 
forms ; nor is it a combination of any of the forms of that 
method. 

If it happens that the sums of money spent on the classes are 
constant, that is, that x^a^ = x^a^, yji^ = y^^^, and so on, the 
formulte reduce to 

Pi xya^a^ + yy^^i^^ + ' 

and consequently also to 

P, ayx^x.^ + ^.yij^y, + ' 

which we know to be correct in these cases, as proved in Chap- 
ter X., the latter being Scrope's method applied to the geo- 
metric means of the mass-quantities. ^ If it happens that the 
mass-quantities purchased of the classes are constant, that is, 
that x^ = 2/1, y„ == 2/p and so on, so that each may be represented 

simply as x, y, •, or that they have all varied alike and x^ = 

ry^, 3/2 = ^^ij and so on, wherefore, the r eliminating itself, 
either may be represented simply as x, y, , the formulse re- 
duce to 

P, xa^ -h ^/?i + ' 

which we know to be correct for these cases, as proved in the 
last Chapter, this being Scrope's method applied to the constant 
mass-quantities. Thus the above complete formulae enclose both 
those sets of formulae, just as the universal conditions to which 
the universal formulae are applicable enclose the two special sets 

^ Also the same reduction will take place if the sums of money are all in the 
same proportion so that x^^^^ = f^x'^'i, ^2^2 — ''l/i'^i) and so on, provided also the 
mass-quantities are in the same proportion, that is, that X2 = rx^, 2/2 = ^2/i, and so 
on ; but then there are no price variations. Otherwise, it is only formulae (6) and 
(8) in Chapt. X. Sect. III. § 3, that are directly applicable in such cases, for- 
mula (8) remaining unchanged (because ^ r eliminates itself from both sides of 

the fraction),. and formula (6) merely reducing to a form in which a, b, , 

representing either all the smaller or all the larger sums spent, take the places of 

a^ and as, bj and bg, But all the other formulas are applicable under the 

proviso that only the sums, and only the mass-quantities, of one of the periods 
be used. Ci'. Note 1 in Ohapt. X. Sect. I. 



A 'rilllM) MHTIIOD 883 

of conditions to which cju'li of those forniiihe was scj)aratcly ap- 
plicable. 

§ 4. Desiring- to get clearer insight into the meaning of this 
fornuila, we may do so hy pntting it in another form, to which 
it easily rednces, as follows, 



•'■- + //■' ■v + ■-■' \} + 



1/^1-^2 , [rj-z , 



('^>) 



The sub-numerators are the total sums of money spent on all 
the goods at the second and at the first periods. The sub-de- 
nominators are the total numbers of mass-units purchased at the 
second and at the first })eriods, that are equivalent over both the 
periods together. Thus the fornuda expresses not an average 
of price variations, but the variation (or constancy) of averages — 
to wit, the variation of the average of the prices at the second 
period of the mass-units then purchased that are equivalent over 
both the periods from the average of the prices at the first period 
of the same mass-units then purchased. These averages of the 
prices are arithmetic averages. But the averages used in ob- 
taining the equivalence of the mass-units are geometric means. 

Thus this method of measuring the constancy or variation of 
the exchange- value of money falls into line — not behind the 
methods adopted by Carli and Young, Jevons, Laspeyres, and 
Messedaglia, and mostly employed hitherto, of averaging merely 
the variations of prices, nor behind the method suggested by 
kScrope, in any of its forms, of comparing the averages of prices 
at each period on the same mass-quantities taken as constant in 
spite of facts to the contrary — but behind the method first dis- 
covered by Drobisch, of comparing the averages of prices at each 
period on the mass-quantities of each period, and so employing 
what we have called double loelght'mg. 

Its resemblance to Drobisch's method — and also its difference 
— is especially plain if we suppose that the labor of finding the 



384 THE UNIVERSAL METHOD 

mass-units equivalent over both the periods has ah-eadv been 
performed, and consequently the numbers of them purchased at 

each period are known. For then, as Vd^'a^' = "^^/?/'/^/' = , 

formula (1) reduces to 

p, - <vy./' + y/'A" + -V + yl' + ' ^ ^ 

which, in form, is the same as the formula for Drobisch's 
method. Drobisch, however, had to use this general formula for 
double weighting on the presupjjosition that the labor of obtain- 
ing the numbers of the mass-units recommended by him had 
already been performed. He was unable to put mto the general 
formula itself his method of selecting the mass-units, or of ob- 
taining the ratio between their numbers. But we have been able 
to do this with our formulae. Our formula, (1), for instance, is 
applicable to the prices and numbers of any mass-units that hap- 
pen to be employed by merchants, as is evident from the fact 
that the larger is the mass-unit employed, the larger will be its 
prices and the smaller its numbers, and conversely, so that every 
full term {x^o.^, or x^>/o.^a.^, etc.) remains unchanged in size what- 
ever be the sizes of the mass-units. What this formula does is 
to reduce, in its second half, the numbers of the mass-units com- 
monly used to the same proportions as are the numbers of the 
mass-units equivalent over both the periods (as shown in Chap- 
ter X. Section III. § 3), so that we are freed from the need of 
knowing either these mass-units themselves or their numbers. 

In the different mass-units used, resulting in different num- 
bers and proportions in the second half of the common formula 
(6), lies the fundamental distinction between the present method 
and Drobisch's. Drobisch sought to draw the average price at 
each period of the mass-units, in all the classes, that are equal 
according to weight (or capacity). This method draws the 
average price at each period of the mass-units, in all the classes, 
that are equal according to exchange-value. Drobisch's method 
used equiponderant mass-units. This method uses equivalent 
mass-units — that is, of course, mass-units that are equivalent 
over both the periods compared. That this method is more 



A THIRD METHOD 385 

nearly right and that method altoo;etlier wronjj, is pUiin from 
this distinction ; for it is plain that not ecjuiponderant, bnt eqniv- 
alent, (or eqnally important) mass-nnits are the economic in- 
dividnals the variations in whose average price we desire to 
measure. 

From this fundamental distinction flow other differences. In 
Drobisch's method different classes can be used at each period, 
no obstacle being offered by it against counting a new class ap- 
pearing at the second period any more than against counting a 
new individual in an old class. In the present method only the 
same classes can be used at each period in the comparison of any 
two periods ; for otherwise a price quotation would be wanting. 
Thus this method must obey the principles laid down in Chapter 
IV. Section V. § 9 ; while Drobisch's method is free from sub- 
jection to those principles. Again, in Drobisch's method we 
have seen a grave defect to be that, even though it obey those 
principles, yet in cases when between two periods there are 
irregular variations in the mass-quantities but no variations in 
any prices whatsoever, it may indicate a variation in the general 
exchange-value of money, thus violating Propositions XXVII. 
and XLIV. ; and also may give two other indications which we 
know to be wrong.^ None of these errors is committed by the 
present method. If no price variations occur, the mass-quan- 
tities may vary as they please, this method indicates only con- 
stancy. Or if all prices vary in the same proportion, the mass- 
quantities may vary as they please, this method indicates only a 
variation of the general level of prices in the same proportion. 
This last statement, which really embraces the first, may be 
proved as follows. Suppose a.^ = ra.^, ^.^ = r/9j, and so on, r 
being the common ratio of variation of every price. (If there 
are no price variations, r = 1.) Then formula (1) becomes 

P2 ^2^"«i + 2/2^ A + x^ayr + y^^yr + 



^ K ^-2«i + 3/2/^1 + ) . ^;;(^-i«i + 3/A + ) _ ^. 

x^a^ + VA + v^r ( x^a^ -\- y^jS^ -f- ) 

« See Chapt. V. Sect. VI. § 5. 
25 



386 THE UNIVERSAL METHOD 

whatever be the variations between x^ and x^, y^ and y^, etc. And 
never will this method indicate a rise of the general level of 
prices, when all prices fall, or conversely. 

§ 5. There is another method which has followed the general 
lines of Drobisch's, — one which is described near the end of 
Appendix C, but which has not yet attracted our attention. 
This is the method invented by Professor Lehr. In this method 
its author has made an eifort to do what appears to be accom- 
plished in the method here presented. He has tried to measure 
the variation in the average price of mass-units, in all the classes, 
that have the same exchange-value over both the periods together, 
— to which equivalent mass-units he has given the not inappro- 
priate name of " pleasure-units." The fault with this method 
lies in the way it measures the equivalence of the pleasure-units. 
Instead of finding mass-units between whose prices at each 
period the simple geometric means are the same, it employs 
mass-units between whose prices at each period the unevenly 
weighted arithmetic averages are the same. No reason is assigned 
for this choice, and it seems to have been made as a matter of 
course. The position is that in a given class, certain different 
sums of money being expended at each of the two periods com- 
pared to purchase certain different numbers of weight-units, the 
total of these sums is expended over both the periods together 
to purchase the total of these numbers ; wherefore a single 
money-unit, on the average over both the periods, purchases a 
number of weight-units represented by the quotient of the total 
sum of money divided by the total of the numbers of weight- 
units, so that we liave here a mass whose average price over 
both the periods is one money-unit. A similar operation is per- 
formed on every class, in each of which a mass is obtained 
whose average price, so measured, over both the periods, is one 
money-unit ; wherefore it is maintained that these masses, being 
equivalent to the money-unit over both the periods, are equal 
pleasure-units. Then the rest of the method is to draw the aver- 
ao-es of the prices of all these pleasure-units of all the classes 
at each period, and to compare them. 

In this position a first error is the use of uneven weighting. 



A THIUl) METHOD 387 

If the ordinary eommcnnal mass-unit of anything'' be priced at 
one money-unit at the first j)eriod and at two money-units at the 
second, and if the ordinary commercial mass-unit of anything 
else be priced at two money-units at the first period and at one 
money-unit at the second, these two mass-units are, over these 
two periods, equivalent mass-units or equal pleasure units, with- 
out regard to the numbers of them that may be purchased at 
either period. The numbers of them purchased at each period 
determine the relative importance of the classes, but not the rela- 
tive importance of the individuals in the classes. A consequence 
of this error is that undue influence may be given to the con- 
ditions existing at one of the periods (or in one of the countries 
whose money is being compared with another's), while, in truth, 
as already pointed out in Chapter IV. (Sect. V.) we ought to be 
esj)ecially careful to allow no greater influence, or weight, to one 
of the periods than to the other. Some deficiencies in the 
method before us following upon the neglect of this princii)le 
will be pointed out further on in this Section and later. 

A second error in this method of obtaining the pleasure-units 
is in the use of the arithmetic average instead of the geometric. 
In such a measurement all the principles examined in Chapter 
YIII. apply. If we prize a mass-unit of [B] twice as highly 
as a mass-unit of [A] at the first period and at the second prize 
the mass-unit of [A] twice as highly as the mass-unit of [B] , 
it is obvious that over these t^vo periods together we })rize the 
two mass-units equally. Therefore the geometric, and not the 
arithmetic, mean is to be used to indicate vsuch ratios of impor- 
tance.** Also in the arithmetic averaging of the prices at the 

■^ Lehr follows Drohisch to the extent of wanting us first to reduce all mass- 
units to the same weight-unit. This is superfluous in his method. Simple in- 
spection of its formula will disclose that the terms in it are unaffected whatever 
be the size of the mass-units employed. It is really a merit in Lehr's method that 
it does not require the use of the same weight-units). — Another merit, where it 
departs from Drobisch's method and agrees with the one here presented, is that 
it requires the use of the same classes at both periods. Wicksell's criticism of 
Lehr's method, noticed in the Appendix, thus strikes at a point in it which 
deserves credit instead of censure. 

8 The case above cited would exist if at the two periods respectively the prices 
are : of A 1.00 and 1.00, of B 2.00 and .50 ; of A 1.00 and 1.50, of B 2.00 and .75 ; 
of A 1.00 and 2.00, of B 2.00 and 1.00 ; of A 1.00 and .50, of B 2.00 and .25 ; or in 
many other combinations. In all these combinations the geometric means are 
equal ; the arithmetic, only in one. 



388 THE UNIVERSAL METHOD 

two periods a variation in the exchange- value of money will de- 
range the result of the calculation. But such a variation has 
no influence upon the geometric mean, as pointed out in Chapter 
IV. ^ It may be added that in using the geometric mean in ob- 
taining our pleasure-units, we escape the imperfections in the 
geometric average with more than two figures, or with uneven 
weighting, which are pointed out in Chapter VIII. (Sect. I. § 6), 
and which have twice troubled us since, and will trouble us 
again presently. For here we are using the geometric mean 
proper, between only two figures, the prices of the same thing 
at two periods, rightly attaching equal importance to each of 
them. Thus what of the geometric method is retained in our 
final formula is flawless. 

Professor Lehr's method has the peculiarity that in consequence 
of its merits and demerits just pointed out, it shares some of the 
defects of Drobisch's method, and escapes others. Thus if no 
prices vary between the two periods. Professor Lehr's method 
always indicates constancy, no matter what be the variations in 
the mass-quantities. But if all the prices vary uniformly, this 
method does not necessarily indicate the corresponding variation 
in the general level of prices unless all the mass-quantities re- 
main constant or also vary uniformly.^" Thus, although, unlike 
Drobisch's method, it respects Propositions XXVII. and XLIV., 
yet, like Drobisch's method, it violates Propositions XVII. and 
XLV. This fact alone would be suflicient to show that there 
is something wrong in it. Yet of all methods hitherto sug- 
gested Professor Lehr's approaches the nearest to the truth in 
theory, if not also in practice. 

^ If we retained this second, but avoided the first error, the formula would be 

P2 ^2^2+y2(^2 + %(«! +"2) +2/1(^1 +<^2)+ 

Pi ajiOi + 2/i/3i + ■a;2(ai + a2)+2/2('3i+^2)+ ' 

This would be better than the method under consideration, but still not true. 

1° If Oo = ^^i, ^2 = ''/^i. and so on, Lehr's formula, which is given in Appendix 
C. VI. § 2, reduces to this, 

/ Xi + rx2 \ n ( y-L+ ry. \ 

P_2 = ^(a;2«i + yai^i 4- ) , ^^°n CTi +a; J + ^^^H 2/1 +2/2 / ^ ' 

Pi a^^a,+y^li^ + /Xi + ra;,N ^ /yi+ry,N ' 

' - ^\ x^^x.J ^- ^V 2/1+3/2 / 
This reduces to unity if r=l; but otherwise it reduces to r only if X2 = sxx, 
2/2 = •^ii and so on. 



A THIRD IMETIIOD 389 

§ (). There is still another way in ^vhich these three methods, 
whieli use double Avcightintr, may be compared. This is by 
comparing the relations of their results in a series of periods. 
In any such series we know with certainty that the results ob- 
tained serially ought to agree with the results obtained directly : 
(1) if the prices have all remained constant through the whole 
series, or have all varied alike at any or every stage, no matter 
what be the changes in the mass-quantities ; (2) if the mass- 
quantities have all remained constant throughout, or have all 
varied alike at any stage, no matter what be the changes in the 
prices ; (3) if at all consecutive periods except two the states of 
things are exactly the same, or the variations are all in the 
same proportion, this having the effect of reducing the irregular 
variations in the series to two sets ; and (4) if the state of things 
at the last period is exactly the same as at the first, or even 
different always in the same proportion, no matter what in- 
tervening changes have occurred. Or even, if we grant Pro- 
fessor Westergaard's position, we may dispense with all these 
restrictions, and say that the agreement ought to take place in 
any and every possible case, no matter what be the changes in 
the prices and in the mass-quantities. 

To begin with a series of three periods : taking the formula 

P, . . 

(1) above reached for p-, and similarly framing the formula for 

1 

P 

p- , and multiplying these by each other, we get the formula for 

the method here presented as it serially indicates the variation 

/ P P P \ 

from the first to the third period ( for p- . p^ = p^ J . This for- 
mula we find to be 

Pa _ 'Vs + 2/»/^3 4- 



( .ryaiaa+ yy/Ji/3a+ ) ( ^■■2^ 02(13+ y'l^"" 1^1^+ ) _ 

(■.rya^2+2/2l//p2+ ) {^'y^3-\-y3^PA+ )' 

But the formula for the direct comparison of the third with the 
first period is 



390 THE UNIVERSAL METHOD 



Pi ajjaj + 2/i/3i 4- ^■3'l^«i«3 + ^/sl^/^A + 

These formulee may agree by chance if it happens that in the 
former the last two thirds yield products in the numerator and 
in the denominator eqnal to, or in the same ratio as, the numer- 
ator and the denominator in the last half of the latter. They will 
regularly agree, as is easily perceived : (1) if a^ = a^ ^ ^v ^^^ ^^ 
on with all the prices, or if a^ = ra^ and a.^ = sa^, and all the 
other price variations be in the same ratios, that is, if there be no 
price variations, or if all the price variations at each stage be in 
the same ratio, no matter what be the variations in the mass- 

quantities I in the former case p^ = 1.00, in the latter p-^ = r s |; 

(2) if .T3 = x^ = x^ and so on, or if x^ = rx^ and x^ = sx^, and 
so on, that is, if there be no variations in the mass-quantities? 
or if all such variations at each stage be in the same ratio, 
no matter what be the price variations ; (3) if «2 = ^p ^^ ^2 ^ ''^p 
and so on, provided either x^ = x^ or x^ = s x^ and so on, or if 
«g = «.„ or «3 = ra.^, and so on, provided either x^ = x^ or x^ = 
sx^ and so on, that is, if there be only one stage with irregular 
variations ; (4) if a^ = a^ and so on, provided either x^ = x^ or 
x^ = sx^ and so on, in which cases both the formulae give unity for 
result, indicating sameness of the price-level, or if a^ = ra^ and 
so on, under either of the same provisos, in which case both the 
formulae give r for result, indicating a general price variation the 
same as all the particular price variations, the conditions here 
being merely that there be no irregular diiferences between the 
first and the last periods, no matter what intervening changes 
may have taken place ; but in no other cases, that is, not univer- 
sally or unconditionally. 

Examining Drobisch's formula in the same way, we find that 
the two measurements universally and unconditionally agree. 
Thus Drobisch's method completely satisfies all these tests. 
This fact we have already noticed ; but we have seen that the 
correctness of Drobisch's method cannot thereby be proved. It 
shares this advantage with other methods clearly false. And it 
fails before other simpler tests. 



A THIRD METHOD 391 

Again, examining Professor Lelir's method in the same way/^ 
we find that tlie two measnrcmcnts agree in cases restricted to 
half of each of the four divisions in which the method here ad- 
vanced is consistent. They agree only (1 ) if «^ = ^^ = o.^ and so 
on, that is, if there be no price variations at all, no matter what 
be the variations in the mass-quantities.; (2) if .1-3 = 'x\^ = x^ and 
so on, that is, if the mass-(][uantities be constant, no matter what 
be the price variations ; (3) if a.^= a^ and so on, provided x^= x^ 
and so on, or if a^ = a^ and so on, provided x^ = cCg and so on, 
that is, if there be only one stage with any variations at all ; (4) 
if «3 = «j and so on, provided x.^ = x^ and so on, that is, if the 
states of things be exactly the same at the last as at the first 
period, no matter what be the intervening changes. It will be 
observed that in every one of the four divisions the proportional 
variations are excluded. 

There is still another possible method using double weight- 
ing, which deserves to be noticed here for the sake of complete- 
ness. This is a form in which Professsor Nicholson's method 
admits of being stated (the third form given in Appendix C, 
VI. §3). A general reason for its not being a successful method 
is tliat it uses even weighting for the inverted variations of the 
mass-quantities. In particular, it falls most abjectly before 
these tests. The agreement between its two measurements de- 
pends entirely upon the behavior of the mass-quantities. It 
takes place only if x^ = x^, or x^ = ^r^ and so on, or if x^ = x^, 
or x^ = /-.P,, and so on, that is, only if there be no irregular varia- 
tions in the mass-quantities at least in one of the stages.'^ 

p, 

1^ Or it may be easier to examine it in conformity witli this formula : ^ = 

"2 

Pi ■ Pi" 

^2 If .^3 =Xi and so on, .to being ditferent, tlie arithmetic average of the vari- 
ations of the mass-quantities (in the last half of the formula) becomes in the 
indirect comparison, a harmonic average of them, lilvcwise with even weighting. 
— The method tliere also (in Note 4) suggested as a variant (with the geometric 
average of the inverted variations of the mass-quantities) has tlie merit that it 
universally satisfies VVestergaard's test ; but it likewise has the defect of using 
even weighting in averaging the inverted variations. If we cured it of the latter 
defect by using uneven weighting adapted for every comparison (as there formu- 
lated in Note 5) it would again lose the former merit. — The method above sug- 
gested in Note 9, submitted to these tests, yields results that agree in exactly the 
same cases as Lehr's method. 



392 THE UNIVEKSAL METHOD 

§ 7. In a series with four or more periods the employment of 
these tests becomes too cumbersome to make it worth while to 
pursue this enquiry into much detail. We must, however, ex- 
amine what happens in such series to the method here advanced. 
The indirect comparison of the fourth period with the first, very 
much abbreviated, is as follows : 

P_4 ^ ^'i"4 + (^1/0102+ ) {xya2a3-\- ) (0:3/03^^4- ) 

^1 ^'i«i+ {-'■Wa^a^-lr ) (.r3V^a2«3+ ) {XiV a^a.^ -\- ), 

while the direct comparison, equally abbreviated, is 

P4_ 3-404 + a-^T/a^g^4- 

i*! ^i^i 4- xya^i-]- " 

These will regularly agree (1) if a^ = a^ = «., = a^ , or, more com- 
prehensively, if a^ = r«g , «3 = s«., and a^ = i^-i j ^^nd so on in 
every case, that is, if there be no price variations, or if all the 
price variations at every stage be in the same ratio, no matter 
what be the variations in the mass-quantities ; (2) if x ^= a^g = 
x.^ = x^ , or if x^ = rx^ , x^ = sx.-, and x^ = tx^ , and so on in every 
case, that is, if there be no variations in the mass-quantities, or 
all such variations at every stage be in the same ratio, no 
matter what be the variations in the prices ; (3) if irregular 
variations take place only at one stage, all the periods then vir- 
tually reducing to two.^^ And it is evident that the series may 
be extended to any length, exactly these relations will hold — 
the two comparisons will regularly agree if any of these condi- 
tions be observed ; but not necessarily in other cases. 

Now among the cases in which these comparisons will not 
necessarily agree is the one in which everything at the last period 
is exactly what it was at the first period — both all the prices and 
all the mass-quantities. In this case we know with absolute 
certainty that the two comparisons ought to agree, no matter 
what be the intervening changes. Our new method, therefore, 
may lead to error. But among the methods we have been re- 
viewing it is only Drobisch's method (along with a few other 
false methods not here noticed ^'*) that still holds out against this 
test. 

^^For Lehr's method, and its modification suggested in Note 9, there are the 
same conditions with exclusion, as before, of the proportional variations. 

^* E. g. the method above alluded to, suggested in Appendix C, V. § 3, Note 4. 



A THIRD METHOD 393 

§ <S. Thus our uicthod fails us twice. It fails even in a series 
of three periods to satisfy Professor Westergaard's full test, 
although in that series it satisfies the certain test yielded by sup- 
posing an exact reversion. But in a longer series it does not 
even satisfy this latter test. It behooves us then to advert to 
this defect in it. 

This defect in the complete method is obviously a survival of 
the defect in the method for constant sums of money, above 
examined at the end of Chapter X. The complete method we 
started out to construct upon each of the partial methods, by 
extending and by modifying them. But it turns out that it 
modifies much more the method for constant mass-quantities 
than the method for constant sums. Now it was precisely the 
method for constant mass-quantities that was perfect ; and this 
has been modified, while the method for constant sums has been 
incorporated whole in half of every formula for our complete 
method. Hence the imperfection in the method for constant 
sums has come over entire into this complete method. 

In that method we discovered the cause of the defect, and a 
way of getting rid of it, though not without loss of other qual- 
ities. The defect can be dispelled by taking the numbers at 
each period of the mass-units that are equivalent over all the 
periods — the pleasure-units, as Professor Lehr calls them, not 
of two periods at a time, but of the whole comprehensive epoch. 
Having already worked these out, we should have the following 
formula for our complete method, expressed in our usual 
symbols,^^ 

P^ a,, -f- b, -h xl" -H yl" + ,^. 

P^-a^ + b7+ •../"+y/" + • ^ ^ 

And now, as all the terms remain the same for every period in 
all the comparisons, the measurements will all agree whether 
made directly between any two distant periods or serially through 
the intervening periods. Here, then, we have an at least con- 
sistent method. 

But the alteration we saw to be impracticable in the case of 
the partial method. It is equally impracticable for the com- 

16 Cf. the formula in Cliapt. X. Sect. V. § 7. 



394 THE UNIVERSAL METHOD 

plete method. We are obliged, therefore, to get along with a 
slightly imperfect form of the complete method — which, how- 
ever, we shall find reason to believe no worse than if it were 
revised in this way. 

In the earlier Chapter we examined the probable amount of 
error incurred by usmg this defective feature in the partial 
method. We found it very small for ordinary cases ; and also 
we detected in the method conflicting and neutralizing tendencies, 
so that when we deal with many classes, and in a long series, 
the jirobabilities are that the results will never deviate to any 
great extent on either side of the truth, but that they will pass 
from the one side to the other of it, always keeping it close 
company, and often coinciding with it. Now the defective 
feature in the whole of that partial method, which here forms 
only half of the complete method, enters into a composition in 
which the other half is without defect. Hence it might seem as 
if the error incurred through that defectiveness ^vould be diluted, 
and lessened by half. Unfortunately this is not so. The de- 
fectiveness of the half leavens the whole. Still the error cannot 
be greater here than there. The error being the same, we need 
not examine it again. 

§ 9. The defect which has been pointed out in this method 
belongs to it in a series of years, the reason for its existence 
there being obvious. This reason for the defect does not touch 
the method used in making a direct comparison between two 
periods — especially between two contiguous periods. The ques- 
tion still remains : In comparing two periods is this a perfect 
method ? — is it the one true method ? The reasoning by which 
it has been reached seems to be faultless. Yet there was a fault 
in one of the premises — in one of the partial methods. And the 
method, even in comparing two periods, is not perfect. We 
have seen it stand certain tests. Unfortunately it does not stand 
all tests. 

One of the tests to which we subjected the partial methods 
previously reached for cases with constant money-sums or with 
constant mass-quantities, was examination as to whether they 
carried out Proposition XXXVL, or not. This, it may be re- 



A TIIIIII) METHOD 395 

called, is the perfectly evkleiit principle that if, making a calcu- 
lation upon all but one or more classes, we find the general 
j)ri(;e variation to be indicated at a certain ratio, and if, later 
noticing the other class or classes, and finding them all to vary 
ill ])rice exactly in this ratio, we insert them in the calculation, 
which we perform over again, the result yielded in the later re- 
calculation ouo-ht to aijree with the result first obtained. Now 
we have seen that the present complete method is composed of 
those two incomplete methods ; and we previously saw that each 
of those methods stood this test, provided its own conditions 
were observed, and not otherwise.'" Those conditions cannot 
exist together (except in absence of all variations) ; and we are 
now investigating cases in which both are supposed to be broken. 
We have seen, also, and see, that that Proposition applies to all 
possible cases, no matter what may be the money-values of the 
classes, or their mass-quantities. Does, then, our present method 
satisfy this test ? It does not. 

That it does not is seen most easily by taking formula (5) and 
treating it in the way supposed. If in all but one class the in- 
dication is of a variation in the general level of prices from 1 
to r, it must be that 



and now if ;% = ij^, it ought still to be that 



\xy^..^ + yyKi% + )' 



V-^2+ 2/2/^2+ +Vi^' 



"^ V xy^^ -\- yyi3j3^ + + 2jyr) 



But this is not necessarily true, — nor is it necessarily true even if 
Z2 = ~i ; nor even if /■ = 1 ; nor even if both these conditions 
occur together (unless the numerators themselves, the total sums 

i« Except in one form of their formulse, common to both, namely Scrope's 
method applied to the geometric means of their mass-quantities, which we have 
called Scrope's emended method, and from which the present method has been 
distinguished. 



396 THE UNIVERSAL METHOD 

spent at each period^ are equal). And the same result is ob- 
tained by using any of the other formultTe for this method.''^ 

Here, then, in this method is a grave theoretical defect. It 
will hardly lead to any inconsistency in practice, since we are 
not apt to make such recalculations, nor are such coincidences 
likely to be found. But the existence of this theoretical defect 
shows that the method, even in a single comparison between two 
contiguous periods, is not perfect. 

This defect exists in all the other methods using double 
weighting — including Drobisch's.^^ It is inherent iu all methods 
using double weighting, no matter which of the three kinds of 
averages of the prices be used. But from it are totally free all 
methods using single weighting (averaging the price variations),^'' 
and Scrope's method in all its forms (this being reducible to the 
preceding). 

Hence in our search after perfection we are throAvn back upon 
these older styles, though upon methods never before employed 
or suggested, and must examine whether they are better. 

III. 

§ 1. That both the geometric method and Scrope's emended 
method carry out Proposition XXXVI. is evident upon simple 
inspection of their formulae. They thus avoid one of the de- 
fects in the method with double weighting above investigated. 
We now need to examine whether they avoid the other defects 
in that method. 

The geometric method does not universally satisfy Professor 
Westergaard's test, as we already know.^ It satisfies it, as is 
easily perceived, in a series of three periods only under three of 
the four full conditions that Ave have seen to be required for the 
method with double weighting, and in a longer series only under 
two of the three there allowed. It satisfies it under the first of 

^'' Its failure iu this respect leads to a defect in this method when extended to 
measuring constancy (or variation) in exchange-value in all things, which will 
be noticed in the next Chapter. 

18 And the method suggested in Appendix C, § 3, Note 4. 

i^Cf. Appendix A, I. §8. 

1 See Chapt. V. Sect. VI. § 7. 



COMPARISON OF THE TIIKEE METHODS 397 

those ooiulitioiis, hocause then the weightinj^ is indifferent ; ^ 
under the tliird, because all methods do so under that condition ; 
and, in the former case, under the fourth, because then the 
weighting is the same in both the comparisons. But it does 
not satisfy it under the second condition, in eitlier case ; for 
here there may be price variations with ditferent weighting in 
the comparisons. Thus this method doe's not behave quite so 
well as that method. Like it, it fails in a long series also in 
the certain case where sameness of the price-level should be 
shown at the end of the serial calculations, when everything at 
the last period returns to what it was at the first. We have 
seen, however, that all such inconsistency would cease, if the 
same weighting be employed through the whole epoch, such 
weighting properly being the geometric average of the full 
money-values of the classes at every one of the periods in the 
epoch. But we have objected to such a procedure as being less 
trustworthy than the use of its own weighting in every com- 
parison of two periods. 

Scrope's emended method fails also before this test, but only 
as did the method with double weighting. In a series of three 
periods, its indirect comparison of the third with the first is as 
follows : 

Pj _ K^ ^2 + /^2^yi3/2 + ) (s^-^g3 + i^yyf, + ) 

Pi {^^yx^x^ + l^yy.y, + ) («ya3,.i-3 -K l^yij^j/, + )' 

while the direct comparison is 

P3 ayx^x^ + I^s^'Ms + 



P 



ayx^x^ + i^y 2/12/3 + 



It is plain that these regularly agree only under all the four full 
conditions above noticed for the method with double weighting. 
These may be briefly recapitulated. They are : (1) if a^ = i-a.^ 
= fsa^ and so on ; (2) if x^ = rx.^ =rsx^ and so on ; (3) if a^ 
= ra^, provided x.^ = sx^, and so on, or if a^ = r«.„ provided x^ 
= sx.,, and so on ; (4) if a.^ = ra^, provided x^ :=sx^, and so on. 
^ See Appendix A, I. § 8. 



398 THE UNIVEESAL METHOD 

And in a series of four or more periods, its indirect comparison 
of the fourth with the first is as follows, abbreviated as before : 

P4_ («X^1^2 + ) (^"^^2^3 + ) iS'-y^z^i + ) 

Pi {a^s/x^^ + ) {a..^Vx^^ + ) (ttg-v/cCgCC^ + ) 

while the direct comparison is 

P^_ ayx^^ + 

"i a^x^^ + 

And again it is evident that these regularly agree only under 
the three conditions noticed for that method, namely (1) if a^ 
= rttg = rsa^ = rsta^ and so on ; (2) if x^ = rx^ = rsx^ = rstx^ 
and so on ; (3) if there be irregular variations only at one stage. 
The series may be extended indefinitely : the formulae and the 
conditions will merely be extensions of these. 

It is plain, again, that also this emended form of Scrope's 
method can be still further emended and cured of this defect, 
if, instead of applying it to the geometric means of the mass- 
quantities at each of the two periods in every comparison, we 
apply it in every comparison to the geometric average of the 
mass-quantities over the whole epoch. 

This is a remedy similar to those we had to invoke for 
the other two methods. To review : The geometric method 
of averaging price variations (with single weighting) can be 
made universally to satisfy Professor Westergaard's test by 
using it with the same weighting over the whole epoch, — say 
of n' periods, — that is, with weights that are the geometric 
averages of the sums spent on every class at every period (e. g. 
V x^a^ ■ x^a^ ■ to n' terms ). The method with double weigh- 
ting above expounded can be made universally to satisfy that test 
by using it with mass-units equivalent over the whole epoch, as 
found by geometrically averaging the prices of the ordinary mass- 
units in every class at every period (e. g. y^o.^ ■ o.^ ■ to n' terms). 

And now the emended form of Scrope's method can be made 
universally to satisfy that test by applying it to mass-quantities 
that are the geometric average of the mass-quantities of every 



COMPARIHON OF THE THKEE MP:TH0DS 399 



class at eveiy period (e. g. '^■^x^^-x.^ ton' terms). It is 

therefore incumbent upon us to examine this system of further 
emending and revising the three methods for the sake of remedy- 
ing one of their defects — in the last its only known defect. 

§ 2. In no case is this remedy satisfactory, for two principal 
reasons: — (1) Because the present epoch is extending every 
year, requiring recalculations ; and it does not appear that a 
later recalculation will be more correct than an earlier. Besides, 
how is a past variation between two years several years ago to 
be affected by present variations ? (2) Because we really do not 
know how to calculate weights, or to determine equivalence of 
mass-units, or to average mass-quantities, over more than two 
periods, since the geometric average loses its virtue when ap- 
plied to more than two figures. Hence it may be that in work- 
ing over these methods into methods universally satisfying 
Professor AVestergaard's test we gain consistency between cross- 
measurements at the expense of other qualities. 

This, in fact, can be shown to be the case with the method 
using double weighting. Suppose states of things over three 
periods as follows : 



I 100 A@ 1.00 100 B@ 1.00 

II 75 A @, 1.50 130 B@ .60 

III 100 A @ 1.00 120 B@ 1.00 



100 for [A] 100 for [B], 
112.50 for [A] 78 for [B], 
100 for [A] 120 for [B]. 



Going from period to period the method with double weighting 

P P P 

gives these results : p^ = 0.9890 and p'^ = 1.0322, whence -^ 

= 1.0208, which is 2.08 per cent, above what we take to be the 
true position. In the direct comparison of the third period with 
the first this method rightly indicates sameness. Now when 
this method is worked over so as to avoid this inconsistency, it 

P P P 

gives these results : p^ = 0.9686 and p^ = 1.0468, whence p- 

= 1.0139 ; and in the direct comparison of the third with the 

P 

first period this method still indicates p^ = 1.0139. Thus, 

though consistent, the revised method twice gives a wrong re- 
sult, bemg 1.39 per cent, too high. 



400 THE UNIVEESAL METHOD 

On this schema the two other methods, gomg from period 
to period, give results very close to those given by the 
method with double weighting similarly used. The geometric 

P P P 

method yields ^ = 0.9902 and ^ = 1.0310, whence ^^ = 

P. 

1.0209. And Scrope's emended method yields -p^ = 0.9885 

P P 

andy = 1.0325, whence ^ = 1.0206. But when they are 

worked over to cover this short epoch of three periods, in avoid- 
ing inconsistency in their final results, they also avoid the 
error incurred by the preceding method. For, so used, the 

P, P 

geometric method yields p^- =0.9698 and ~ = 1.0311, whence 

P 

~ = 1 .00, which is also indicated in the direct comparison ; 

1 

P P 

and Scrope's emended method yields ^ = 0.9953 and p^ = 

1 2 

P 

1.0046, whence p^ = 1.00, which is also indicated in the di- 

rect comparison. 

The question, however, arises : Is Professor Westergaard's 
test correct universally ? The case before us is of such a nature 
as to throw doubt ujdou it. Here the prices of both the classes 
and the mass-quantity of [A] alone have reverted at the third 
period to what they were at the first. Had this third period, 
with the sameness of its prices, immediately followed upon the 
first period, there would be no question but that the exchange- 
value of money is constant (in accordance with Proposition 
XLIV.). But an intervening period has separated the two ; 
and now, while the same number of economic individuals in [A] 
fall in price during the second stage as rose during the first, in 
[B] a greater number of such individuals rose in price during 
the second stage than fell during the first. Had the mass-quan- 
tity of [B] fallen back at the third period to what it was at the 
first, there could again be no question but that the level of 
prices at the third period has returned to what it was at the first, 



COMPARISON OF THE THREE METHODS 401 

— whicih would be iudicuted by all these, and by several other, 
methods. But ought not the fact that, while the changes in [A] 
counterbalance each other, a greater number of individuals in 
[B] have risen than fallen, be allowed to show that the level of 
prices has risen more than it has fallen, so that it is rightly 
placed by all our three methods, in their serial use, slightly 
above its first position ? This would involve also that the ex- 
change-value of money has fallen somewhat, — which, seeing that 
prices are exactly the same, is somewhat hard to entertain. 

Yet another example casts doubt upon Professor Wester- 
gaard's universal test. Suppose that both prices and mass- 
quantities vary irregularly between a first and a second period. 
And suppose that between the second and a third period there 
is irregular variation of mass-quantities, but no variation of 
prices. There is, then, no variation of the general price level 
between these last periods. Therefore, the indirect comparison 
of the third with the first period will show the same general 
price variation as between the second and the first. But the 
direct comparison of the third with the first will show a differ- 
ent price variation from that between the second and the first, 
and consequently from that indirectly obtained between the 
third and the first. Now of these two measurements the latter 
has more reason in its fiivor. 

Still, even if we should deny Professor Westergaard's test 
in such cases, we should gain little comfort in regard to onr 
methods, since there is one case in which his test is perfectly 
certain, and which none of the methods (except in their doubt- 
ful revised forms) can satisfy. This is when at any later period 
the prices and the mass-quantities both revert exactly (or pro- 
portionally) to what they were at some earlier period. This is 
a test which no sound method yet devised or suggested, in 
going from period to period over all the intervening periods, 
will stand. 

That is, none of the known methods that hold out against 

other tests will stand this test in the world such as we have it 

— with varying mass^-quantities as well as with varying prices. 

But if we had an economic world, — or supposed one, — in which 

26 



402 THE UNIVERSAL METHOD 

forever the same mass-quantities, or mass-quantities in the same 
proportions, are bought and sold, at prices varying according 
to demand only, then both Scrope's emended method and the 
method with double weighting above described — both of which 
are in these cases the same as Scrope's method applied to the 
constant mass-quantities, — would completely and absolutely sat- 
isfy Professor Westergaard's test, and all other tests. In such 
a world, the argument for this method being convincing, we can 
be certain that we have the absolutely true method of measur- 
ing variations in the exchange-value of money. But in the 
world as it is, we have not yet reached the absolutely true 
method. 

§ 3. We can, however, be sure that we have come pretty 
near to it. 

In the first place, we have three distinct methods, for each of 
which much can be said, which in some cases regularly give the 
same results, and which in all ordinary cases give results very 
close together. The cases when they exactly agree are when 
we are dealing with two classes equally important over both 
the periods. 

In these cases we may first prove that the geometric method 
(which now must use even weighting) exactly agrees with the 
method with double weighting above expounded. The con- 
dition to be observed is that the two classes are such that 



>/x^a^x/j.^ = "^2/1/^13/2/^2 } ^^> which is the same thing, x^a^x^a^ = 
Vi^iVj^i • Fi'om this condition is derived 



^2'h y2^2 •'»2«2 + 2/2/^: 



vA -v^i ^i^^-i + yA 

Therefore in the first half of formula (1) for the method with 
double weighting, applied to two such classes, we may substitute 
one of these values of that half, and proceed to reduce thus, 

yA _ a-y^///^ + yyi^A ^ 1^2 G^i3/2^ss + 2/12/2^/^ A ) 
«^i«i ^'2^«i«2 + y2^^A «1 (a^ia^2^«i«2 + ^iy2^f^A ^ 

B 5 
From the given condition is also obtained x^x^ = y^y^ ^-^ , and 

a^«2 



COMPARISON OF THE THREE METHODS 403 



2/1^2 = ^'1**2 q 0' Therefore the last expression becomes 

^ .(^.yy<^. + ^.^2^ j jy<^^^. (-.y. + -^i'^ 

V^«l«2 ^ ^ ^''V^2 ^ 

Agam from the given condition is obtained ^,^=---, and 

' ^ = '— . Therefore the last expression becomes 
«i«2 yuh 

«y/5A(^»'i»'22/i2/2 + ^^12/2) «i'^;V^2 ^«i /'^/ 

which last is the formula for the geometric average of the price 
variations with even weighting. Q. E. D. 

Next we may prove that in these cases Scrope's emended 
method likewise exactly agrees with the geometric method. 

From the given condition is derived \/x^x.^^ V^i^2"^^j ^^^ 

^ ''•12 

on inserting this value of s^x^x.^ in the formula for Scrope's 

emended method confined to the two classes, and reducing, we 
get 

P2 ^ '^-^ ^yiy/ ^ii ^ + ^^2^yiy2«iS 
^1 '^yyiyJ^J^t + f^yyxy-/h% 

_ ^"-A ( ^«A + ^^^ 2) _ K ^2 Q E D 

Thus, when we deal with hvo classes equally important over both 
the periods, the method with double iDei</]ding, and Scrope's emended 
method, and the e/eometric method, all yield the same result. 

In all such cases (which permit both the money-sums and 
the mass-quantities to be different at the two periods, but re- 
quire them to yield equal products) the three methods satisfy all 
our tests. We may be sure, then, that in these restricted cases 
the common result is the true one. 



404 THE UNIVERSAL METHOD 

§ 4. In other ordinary cases trial shows that in their resnlts 
the three methods do not diverge considerably from one another. 
Therefore we have reason to believe that they do not deviate 
considerably from the truth. 

As regards their divergence amongst themselves, trial seems 
to show that, only two unequal classes being employed, the 
method with double weighting generally gives a middlmg re- 
sult. The highest is given by the geometric method and the 
lowest by Scrope's emended method, when the preponderating 
class is with price rising above the general average ; and re- 
versely the lowest by the geometric method and the highest by 
Scrope's emended method, when the preponderating class is 
with price falling below the general average. With more classes 
this rule does not seem to hold, unless most of the larger ones 
have prices varying in the one direction and most of the smaller 
ones have prices varying in the opposite direction (in relation to 
the general average price variation). The greatness of the di- 
vergence is determined both by the greatness of the inequality 
in the sizes of the classes and by the greatness of the price 
variations — principally by the latter. It is greatest when the pre- 
ponderating class varies little and the smaller class varies much ; 
but it may also be considerable when the preponderating class 
varies much and the smaller class varies little, provided the 
preponderance is not excessive ; for when it is excessive all the 
results are drawn so far with the variation of the excessively 
preponderating class that their divergence may be lessened al- 
most to nothing, — and such, of course, is the event also in cases 
where the classes, in pairs or sets, are nearly equal in size. In 
the cases of moderate inequality in the sizes of the classes and 
of excessive variation in one of the prices, there seems to be a 
1:endency on the part of the geometric method to deviate by 
itself, becoming untrustworthy, while the other two methods 
keep fairly close together. All this about the geometric method 
in general agrees with what has already been found to be the 
deviation of this method, in the partial cases, in compari- 
son with the component parts of the method with double 
weighting. 



COMPARISON OF TIIK THREE METHODS 405 

The followiiit"^ examples, purposely extravagant, will illustrate 
some of the salient positions. 

I 10 A @ 1.00 50 B @ 1.00 I 10 for [A] 50 for [E], 
II 6A@1.50 70B(^r,), .40 I 9 for [A] 28 for [B]. 

[B] is O.04 times larger than [A]. The geometric method 

indicates ^ = ().522B ; the method witli double weighting makes 

' 1 
it 0.5240, and Scrope's emended method, 0.5273. 

I 100 A @ 1.00 30 B @ 1.00 I 100 for [A] 30 for [B], 
II 80 A @ 1.50 40 B@, .20 I 120 for [A] 8 for [B]. 

[A] is 7.071 times larger than [B]. The indications, in the 
same order, are 1.108(J, 1.1547, and 1.1370. 



I 100 A @ 1.00 3 B @ 1.00 
II 80 A @ 1.50 4B@ .20 



100 for [A] 3 for[B], 
120 for [A] 0.80 for [B]. 



[A] is 70.71 times larger than [B]. The indications are 
1.4585, 1.4555, and 1.4515. 



I 100 A @ 1.00 300 B @ 1.00 
II 80 A @ 1.50 400 B@ .20 



100 for [A] 300 for [B], 
120 for [A] 80 for [B]. 



[B] is 1.414 times larger than [A]. The indications are 
0.4608, 0.4634, and 0.4667. 



I 100 A @ 1.00 50 B @ 1.00 
II 80 A @ 10.00 60 B @ .90 



100 for [A] 50 for [B], 
800 for [A] 54 for [B]. 



[A] is 17.21 times larger than [B]. The indications are 
9.4630, 6.6824, and 6.5369. 



I 100 A @ 1.00 100 B @ 1.00 
II 98 A @ 2.00 102 B@ .10 



100 for [A] 100 for [B], 
196 for [A] 10.20 for [B]. 



[A] is 4.38 times larger than [B] . This example may be com- 
pared with one in Chapter XI. Section IV. § 3. The mass- 
quantities have varied so little that it almost comes under the 
cases with constant mass-quantities. If it did so, the mass- 
quantities of the first period remaining constant, we know with 

certainty that the price variation would be ,y = 1.05. But the 

variations of the mass-quantities, slight as they are, make [B] 
larger relatively to [A] than it otherwise would be, and so give 
greater eifect t<> the fall of })rice. Therefore we know with cer- 



406 THE UNIVERSAL METHOD 

tainty that the indication should be slightly below 1.05. Now 
the method with double weighting yields 1.0442, and Scrope's 
emended method 1.0405, while the geometric method yields 
1.1464, thus being certainly wrong. Which of the former two 
results is nearer the truth, it is impossible to tell. That the 
geometric method should fail in such an extravagant example, 
ought not to be counted much to its discredit. All these ex- 
amples being extraordinary in their variations, the general 
closeness of the results yielded in such cases by the three methods 
is a warranty of their greater closeness in all ordinary cases. 

§ 5. In the second place, what has been shown of the geo- 
metric method with reference to the partial method for constant 
sums, and of that method with reference to the truth, evidently 
belongs to the geometric method in general, and to the method 
with double weighting, which contains that partial method, and 
also to Scrope's emended method, which likewise comprehends 
that method. That is, there is neutralization both between the 
many classes that are measured together and between the suc- 
cessive periods in a series. In consequence of this last quality, 
even though a considerable error should be made at one stage, 
there is probability of its being corrected at another, and there 
is little likelihood of the error being any greater at the end of 
a long series than near the beginning — except in case of con- 
tinual tendency of the level of prices in one direction, although 
even in this case there may be neutralization through changes 
in the sizes of the classes. A complex example illustrative of 
some of these inductions will be given later. 

Lastly, the amount of the errors at any later period may be 
subjected to a certain test, which generally shows but slight 
deviation — although this test is not as satisfactory as we might 
desire. This test is to suppose the period in question to be fol- 
lowed by a period with everything exactly the same as at some 
earlier period, and then to calculate on to it, to see how the re- 
sult serially reached for it compares with unity, which is known 
to be the true result. This, however, is by no means a perfect 
test, for two reasons. The one is that the error proved for the 
supposed period is not necessarily the error for the other periods 



COMPARISON OF THE THREE METHODS 407 

which preceded, nor a definite increase upon their errors, whence 
their (>rrors may he calcuUitcd. For there is no gradual ac- 
cuniuhition of error, hut irregularity, and some of the preceding 
results may be above, and some below, the truth. The other 
reason is that this last calculation is nothing l)ut the inverse of 
a direct calculation from the earlier period to the last actual 
period ; but such a direct calculation we know to have no 
greater validity than the indirect calculation. Still, the fact 
that in practice the direct and the indirect comparisons do not 
diverge much — and especially the fact that they do not diverge 
more at the end of a long series than near the beginning, is 
cood evidence that none of the methods deviate much from the 
truth. 

§ 6. Between the three methods our choice may be guided by 
what we have so far learnt. The geometric averaging of the 
price variations, with single weighting' is probably tlie least 
trustworthy, because we have seen that the geometric average 
between more than two equally important things is not to be 
depended upon — and we have sometimes caught it flagrante 
delicto. The method with double weighting, using the geomet- 
rically measured equivalent mass-units, has been led up to by 
a chain of reasoning which seems to be sound. Yet there was 
something defective in the reasoning at an early stage, since 
even the partial method for the cases with constant sums was 
found to fail before some of the tests. For the form of 
Scrope's method, in which it is applied to the geometric means 
of the mass-quantities, the argument is^that this method under- 
lies both the partial methods, and so is the one fundamental 
method, which, being true in those two'cases, or at all events in 
one of them, ought to be, if not absolutely true, yet near to the 
truth, in all other cases. But the decisive argument for it is 
that, while it stands all the tests that are satisfied by the 
method with double weighting, it stands still another test, be- 
fore which that method fiiils. Hence this method (although 
the geometric method shares with it this last quality) is, in all 
probability, the best of the three. 

§ 7. The fact that we have not reached a perfectly certain 



408 THE UNIVERSAL METHOD 

method — except for the cases when the mass-quantities (the 
physical parts of the economic world) are constant, or all vary 
alike, and the very special cases when there are only two equally 
important classes, or pairs or sets of such classes, not to mention 
the cases when the prices are constant, or all vary alike — must 
not be misinterpreted. It does not mean that in cases when the 
mass-quantities and the prices irregularly vary, there is no one 
true variation of money in general exchange- value ; for if that 
were so, there could, in these most common cases, be no varia- 
tion of money in general exchange-value at all, which is absurd. 
What it means is that our mathematics, so far as yet carried in 
the subject of averaging, fail us. We have not yet found the 
right average — or the right weighting for averages already 
known. It may be there is no average that is perfectly cor- 
rect — or no weighting that will make it so. Perhaps no 
method exists to be found that will stand all our tests. But 
from the fact that the perfect average, and the perfect weight- 
ing — or the perfect method of combining them — have not yet 
been discovered, it does not necessarily follow that they are 
never to be discovered. Or if finally we must abandon the 
search as hopeless and believe that no perfect method exists to 
be discovered, this failure of mathematics would not disprove 
the existence of one true variation. The fact, however, that we 
have three methods — not to mention two or three more, as will 
be shown presently, — which in all ordinary cases give results 
very close together, and which we have every reason to believe 
to be close to the truth, and to hold the truth between them, 
ought to make us fairly content.^ 

5 Here may be inserted a suggestion of a line along which it might be thought 
that mathematicians may perhaps be able finally to solve our problem with pre- 
cision, and at the same time a warning against over-expectancy. When the mass- 
quantities are constant, we have seen that the solution is perfect. Such cases 
may, then, be used as a touchstone for the rest. Now let mathematicians find 
the weighting — according to means of some sort between the full money-values at 
each of the two periods compared — which will make the geometric average of the 
price variations always agree with Scrope's method applied to the constant mass- 
quantities. If this task be accomplished, it might seem as if the geometric aver- 
age of the price variations with the same kind of weighting would universally be 
correct, including the cases when the mass-quantities vary. An approach toward 
this solution may be indicated. By trial it is found, at least with two uneven 
classes, that the geometric average of the price variations with weighting accord- 



OTHER METHODS EXAMINED 409 

IV. 

§ 1, It will be well also to examine other methods, but es- 
pecially the convenient form of Scrope's method in which it is 
applied to the arithinetic means of the mass-quantities at each 
period, or, which is the same thing still more conveniently, to 
the aggregates of the mass-quantities at both periods. 

Submitting this method to Professor "Westergaard's test, we 
have for the indirect comparison 



P^ a^{x, + x^) + /3j(2/^ -f 2/2) -h ■ 

«3(»^2 + »^3) + I%bj2 + ^3) + 



sK + ^-i) + t%{y2 + 2/3) + 



ing to the arithmetic means of the full money- values at each period gives results 
with error opposite to the error given by the geometric average with geometric 
weighting. For example : 

100 A (5) 1.00 100 B @ 1.00 I 100 for [A] 100 for [B] - 200, 
100 A @ 1.96 100 B @ .09 | 196 for [A] 9 for [I?] - 205. 

P 205 
The known result is p^ = ^ = 1.025. The geometric weighting is 4.666 for [A] 

to 1 for [B]; and with this weighting the geometric average of the price variations 
is 1.1426. The arithmetic weighting is 2.715 for [A] to 1 for [B]; and with this 
weighting the geometric average of the price variations is 0.8552. It is plain that 
the proper weighting must be given by some mean lying between the geometric 
and the arithmetic. One such mean has been discovered by Gauss, who named 
it the arithnietico-geometric mean. If the classes be weighted according to this 
mean, the weighting M'ill be 3.657 for [A] to 1 for [B]; and with this weight- 
ing the geometric average of the price variations is 1.014, which is slightly too 
low. Other examples likewise show that the geometric average with geometrico- 
arithmetic weighting comes nearer to the truth than the geometric average with 
geometric weighting ; but that it still errs on the side of the geometric average 
with arithmetic weighting. In our example the proper weighting for the geo- 
metric average — that which will make it give the true answer — is 3.7526. A still 
closer approach to this is made by a mean which is a combination of all the three 
common means, and which may be called the arithmetico-geometrico-harmonic 
mean. This is the harmonic mean between the arithmetic and geometric means 

between the two quantities in question, the formula for which is 7:=^ . 

a + b + 2yab 
Weighting the classes according to this mean between their two full money-values, 
we find the weight for [A] to be 3.71S times that for [B]. With this weighting 
the geometric average of the price variations is 1.0201. In practice, however, 
this method would hardly differ from the preceding, which would differ very 
slightly from the completely geometric method. There is, moreover, a consider- 
ation which invalidates the idea that a mean perfectly good for our purpose exists : 
for if the money-values of all the classes be constant, all kinds of means are the 
same, and all are then defective, as proved in the next preceding Chapter on the 
method for constant sums. 



410 THE UNIVERSAL METHOD 

while the direct comparison is 

P3 a^x^ + x^ + ii^{y, + 2/3) + 



P^ a^{x^ + X3) + /3^(2/i + 3/3) + 



These regularly agree : (1) if o.^ = a^ = «i, and so on, whatever 
be the variations in the mass-quantities ; (2) if x^^ x.^^= x^, and 
so on, whatever be the variations in the prices ; (3) if a^ = a^, 
provided x^ = x.-,, and so on, or if «., = a^, provided x,^ = x^, and 
so on ; (4) if a^ = «j, provided x^ = x^, and so on. These are 
the four divisions in which Scrope's method applied to the geo- 
metric means and the method with double weighting regularly 
agree ; but in each division only half of the conditions are re- 
tained, uniform variations in the prices or in the mass-quan- 
tities being cut off. In a series of four or more periods it is 
easily seen that there will be necessary agreement only (1) if 
there are no variations in the prices, or (2) if there are no varia- 
tions in the mass-quantities, or (3) if the periods fall into two 
sets in each of which there are no variations at all. In short, 
this method behaves in this matter exactly like Professor Lehr's 
method. 

§ 2. Still other limitations may be discovered in this method, — 
by which limitations also Professor Lehr's method will be found 
to be restricted. The investigation may be opened with examina- 
tion as to what is the relation between the arithmetic means, or 
between the aggregates, of the mass-quantities of the different 
classes in cases when the result is known with certainty. These 
are when two classes are dealt Avith that are equally important 
over both periods. For simplicity we shall begin with cases 
where the result is unity, indicating constancy in the exchange- 
value of money ; and always we shall follow the usual practice 
of employing mass-units that are equivalent at the first period. 
The following is the simplest schema : 

I 100 A @ 1.00 200 B @ 1.00 I 100 for [A] 200 for [B] — 300, 
II 100 A @, 2.00 200 B @ .50 | 200 for [A] 100 for [B] —300. 

200 400 

Here the mean, or aggregate, mass-quantities purchased over 
both the periods together are twice as many of these mass-units 



OTHER METHODS EXAMINED 



411 



of [B] as of [A] . This is in accordance with the formula dis- 
covered in Chapter IX. Section II. § 5, namely, 



/ = 



.'« - 1) 



for here 



y' = 



^-'(2 - 1) 



^-k 



= 2.r 



Now the same result, constancy, appears also if things hap- 
pened as represented in the following schemata : 



and 



I 100 A @ 1.00 100 B @ 1.00 
II 50 A @ 2.00 200 B @ .50 

150 300 



I 100 A @ 1.00 150 B @ 1.00 
II 75 A @ 2.00 200 B @ .50 



and 



175 



350 



I 100 A @ 1.00 300 B @ 1.00 
II 150 A @ 2.00 200 B @ .50 

250 500 



100 for [A] 100 for [B] — 200, 
100 for [A] 100 for [B] — 200; 



100 for [A] 150 for [B] — 250, 
150 for [A] 100 for [B] — 250; 



100 for [ A] 300 for [ B] — 400, 
300 for [A] 100 for [B] — 400; 



and an indefinite number more of such exam])les might be made. 
In all these it is observable that, while the states of things per- 
mit of even weighting, aud the price variations are geometric, 
wherefore the result is known to be constancy, with the further 
condition that the total sums of money spent on both the classes 
together at each period are the same, the aggregates, and hence 
the arithmetic means, of the mass-quantities of each class pur- 
chased over both the periods together are twice as many of [B] 
as of [A] . This constant relation of two A's to one B (these 
being equivalent at the first period) is comparable with the rise 
of A in price from one to two money-units. Now under these 
conditions this relation between the arithmetic means, or ag- 
gregates, of the mass-quantities of each class over both the 
periods may be proved to be universal. The general schema is 



I x/ A @ 1 y'B@l I x/ for [A] y/ for [B] — 3\' + y/, 

II x/ A @ «/ y/ B @ /?/ I z/a./ for [A] y/^/ for [B] — x/a/ + 2///3/; 



412 THE UNIVEESAL METHOD 

in which the conditions, beside the equivalence of the mass- 
units at the first period, are : (1) x^x.^a^ = y-^'y^^^' , (2) ^^ 

=z — J J (3) x.-,'o..J + 2/2' /V = »/ + y^ • We wish to prove that 
From the first and second conditions is derived 



I ^, I 



. ^ VMK = J/IL, , whence a - - ^^^^ 



and the third condition reduces thus, 



/ 
2 



2 

»'2'«2'' + 3/2' = «2'(^/ + 2//). 

2/2' (2// + «^/) = ^'1<{^I + 2/1O; 
2/2' = ^'l'^- 

T* n(* o 
From the first condition is derived y^ = ^ /^ / ; wherefi^re 

2/2 r 2 



Therefore the sums of these equals are equal. Q. E. D. This 
shows, incidentally, that our previous formula, which with the 
relation between the prices above supposed in the second condi- 
tion reduces as follows, 

is true, not only for the purpose for which it was invented, but 
also, with y' = y/ -|- y.^ and x' = iw/ -|- 3/3', for the purpose 
in which we are at present interested. Thus in our particular 
examples, where a^ = 2.00, the arithmetic mean number of B's 



OTHER Mf^THODS EXAMINED 413 

purchased over both the periods will always be twice the mean 
number of A's, when these conditions are fulfilled. And on 
these relative mass-quantities Scrope's method always indicates 
constancy ; for 



P2 _ 1 X 2 + 2 X ^ _ 2+ 1 
Pi~lxl + 2xl~"l + 2 



= 1.00. 



And so in the universal cases : applied to the universal rela- 
tions between the mass-quantities, when the above conditions 
are fulfilled, Scrope's method always indicates constancy, and 
therefore always yields the right result ; for 

P, 1 X «/ + «/ X /9/ «/ + 1 

P, 7 1 X 1 + < X 1 ~ 1 + < ~ • ' • 

Now, if we used other mass-units, we should get different re- 
lations between their aggregate numbers in the two classes, but 
the new aggregate numbers would be equally constant. In 
general we know that the results obtained by Scrope's method 
are the same whatever be the mass-units used. Therefore, 
what has above been proved of Scrope's method using certain 
mass-units may be universalized of it using any mass-units, and 
we have this general proposition : When the conditions are such 
that, with two classes, their prices varying from unity to the op- 
posite geometric extremes, the weighting is even over both the pe- 
riods, wherefore we hiow that the geometric mean of the price 
variations is true, and it here indicates constancy, then, provided 
fur^ther that the total suins spent on the two classes together at each 
period are the same, also Scrope's method applied to the arithmetic 
means {or to the aggregates) of the mass-units of each class pur- 
chased at each period is always correct. It will be noticed that 
the cases are extremely limited in which Scrope's method, so used, 
can be exactly right, except by chance. 

§ 3. Let us now widen the restriction by leaving off the last 
condition. We may take another numerical example, with the 
same price variations as in the preceding, but with different 
mass-quantities. Things might happen as represented in this 
schema : 



414 



THE UNIVERSAL METHOD 



I 100 A @, 1.00 80 B @ 1.00 I 100 for [A] 80 for [B] — 180, 
II 88 A @ 2.00 440 B @, .50 | 176 for [A] 220 for [B] — 396. 



188 



520 



Here we likewise have even weighting over both the periods 
(as 100 X 176 = 88 X 220), so that we knoAV that the geo- 
metric mean of the price variations is correct, and this still indi- 
cates constancy, — as is also indicated by Scrope's method itself, 
applied to the geometric means of the mass-quantities. But the 
arithmetic means (or the aggregates) of the mass-quantities of 
each class over both the periods are no longer in the relation of 
2 B's to 1 A. And now Scrope's method, applied to the arith- 
metic means of these mass-quantities, thus 



94 X 2 -f 260 X i-_318 
94 X 1 -h 260 x"l ^ 354 



= 0.8983, 



indicates a fall of 10.17 per cent., which is far astray. 

But the above numerical schema may be analyzed into the fol- 
lowing : 



I 100 A @ 1.00 80 B @ 1.00 

'40 A @ 2.00 200 B@ .50 

II <! 40 A @ 2.00 200 B@ .50 

8A@2.00 40 B@ .50 



100 for [A] 80 for [B] — 180, 

80 for [A] 100 for [B] — 180 1 

80 for [A] 100 for [B] — 180 

16 for [A] 20 for [B] — 36 J 



Here we see that at the second period after we have spent the 
same total sum of money as at the first — 180 money-units (but 
in money, notice, that is shown to have the same exchange- 
value at both the periods), — the fact that we go on spending 
more money on the two classes in the same proportions in no 
wise aifects the exchange-value of our money at this period. 
Hence all that we are concerned with in the comparison is ex- 
pressed m the following : 



I 100 A 
II 40 A 

140 



1.00 80 B 
2.00 200 B 

280 



1.00 
.50 



100 for [A] 
80 for [A] 



80for[B]— 180, 
100 for [B]— 180, 



And here we again have the same relations that we had in our 
earlier examples, the arithmetic mean (or aggregate) number of 
B's purchased over both the periods being twice that of the A's. 



OTHER METHODS EXAMINED 415 

Therefore Scrope's method applied to the arithmetic means (or to 
the aggregates) of these numbers of mass-units gives the true re- 
sults — and only when applied to these. 

That Scrope's method, so applied, is always correct when 
these conditions are fulfilled, may also be demonstrated. The 
universal schema is the one above used ; but now the conditions 
are only the former two, namely (1) x/x./a^' = y^'y^ fi.^ , and 

(2) [i.^ = y. Now if we reduce the total sum spent on both 

the classes at the second period, namely x^aj -\- y^^^ ^ to what 
is spent on them at the first period, namely x/ + y.^ , we may 
do so by reducing the particular sums spent on [A] and on [B] 
at the second period in the same proportion, and consequently 
also the numbers of the mass-units purchased of [A] and of 
[B] at the second period likewise in the same proportion ; and 
doing this, we get the following, schema : 

I a-/ A @ 1 2// B @ 1 

II ^^,^^^^,^ A @ a/ y{^^/ + ^;;), B @ /?/ 

I ;r/ for [A] 2// for [B], 

From the given conditions we derive x/x./a./ = y(y.l , and 

"■•I 

= x/Xj' ; and by reducing the expressions for the sums spent on 
[A] and [B] respectively at the second period, and by substi- 
tuting these values, the former reduces to y^' , and the latter to 
x/ ; wherefore simpler expressions for the numbers of A's and 

y' 
B's purchased at the second period are — y or y^' [i.^ for the A's 

X ' 
and ~ or x^a^ for the B's, and the schema becomes 

r2 

I a-/ A @ 1 2// B @ 1 I a;/ for [A] 2// for [B], 

II ^ A @ < x/«/ B @ /3/ 1 2/x^ for [A] .r/ for [B] ; 

"2 1 

and the total of the numbers of A's purchased over both the 



416 THE UNIVERSAL METHOD 

periods is -^^—^ — , , and that of the numbers of B's is 

x^'a^ + 2//j showmg that the B's are a^ times the A's. Now 
Scrope's method applied to these relative total mass-quantities, 
and therefore also if applied to the arithmetic means, always in- 
dicates the right answer, constancy, as above shown. Q. E. D. 

Thus we find that in all cases where, with two classes equally 
important over both the periods, the exchange-value of money 
remains constant, a condition for Scrope's method applied to 
the arithmetic mean of the mass-quantities being correct is that 
we must make use only of those mass-quantities which are to- 
gether purchased at each period with the same total sum of 
money/ Hence we may induct that in all cases, when the 
exchange-value of money remains constant, a condition of 
Scrope's method, so applied, being correct, is that we must 
apply it only to the mass-quantities which we have so reduced. 
Thus a necessary preliminary worh for the employment, with ex- 
pectation of the best results, of Scrope^s method applied to the 
arithmetic means of the mass-quantities, is that we must reduce the 
mass-quantities to those ivhich are purchased at both periods with 
the same total sum of money (^provided this be constant in ex- 
change-value). 

Of this another brief proof may be supplied by the use of a 
test case. Suppose these states of things : 

100 for [A] 100 for [B]— 200. 
112.50 for [A] 78 for [B] — 190.50, 
50 for [A] 50 for [B] — 100. 

Here it is evident that money has the same exchange-value at 
the third as at the first period, and this ought to be indicated by 

1 The same reduction is required, in these cases, also if we should use Scrope's 
method applied to the smaller mass-quantities at either period, that is, to the 
mass-quantities common to both the periods. Then, under the conditions above 
supposed, this method likewise gives the right results ; for, supposing Oj' > 1 ^^^ 

Bo' < 1, the smaller mass-quantities are —. A and 2/i'B, that is, Cg' times more B's 

than A's, as before. — Also, in the same cases, the geometric average of the price 
variations with the weighting of the smaller total exchange-value of each class at 
either period will always give the right result ; for the smaller total money- values 
are 2/1' for each class, that is, the same, so that even weighting will have to be 
used. Thus the same reduction is required here also. 



I 100 A @ 1.00 100 B @ 1.00 

II 75 A @ 1.50 130 B @ .60 

III 50 A @ 1.00 50 B @ 1.00 



OTIIKi; MKTHODS EXAMINED 417 

the two results obtained for the two sets of price variations. Our 
three superior methods give the correct fiual answer ; for the 

P P, 

geometric method indicates n= 0.9902 and p' = 1.0098, the 

' P, ' P 

method with double weighting p^ = 0.9890 and p- = 1.0111^ 

1 2 

P, P. 

and Scrope's emended method ,," = 0.9885 and p'=1.0116y 

1 2 

P 

whence in every instance p^ = 1.00. Not so Scrope's method 

applied to the arithmetic means of the mass-quantities ; for its 

P P P 

results are p' = 0.9888 and / = 1.0821,whevicep*= 1.0206, 

^1 ^1 ^1 

wrongly indicating a rise of 2.06 per cent.^ But if all the 

mass-quantities are reduced to those which, in the same propor- 
tions as actually purchased in, were purchasable at every period 
with the same total sum of money, applied to their arithmetic 

P, P. 

means Scrope's method yields p- = 0.9922 and ~ = 1.0078, 

^ I 1 

P 
whence p^r= 1.00.^ 

This example is one in which Professor Westergaard's test has 
been shown to be satisfied by our three methods (all containing a 
geometric element), but not by any of the purely arithmetic 
methods — Professor Lehr's included, — illustrating condition (4) 
above in § o of Section II. and at the commencement of this Sec- 
tion. It is plain, then, that also Professor Lehr's method requires 
a similar reduction of the mass-quantities before it can bring 
out its best results. In the example just given this method era- 
ployed in the direct comparison of the third with the first period 
shows sameness of exchange-value of money at the third as at 
the first. But employed in measuring each variation separately, 

2 Applied to the smaller mass-quantities at either period in each comparison, 
Scrope's method is still more wrong in these cases ; for then its indications are 

^ = 0.9857, :^ = 0.9523, and ^ - 0.9387. 

3 Applied, as before, to the smaller of these reduced mass-quantities at either 
period, Scrope's method again gives the right final result ; for its intermediate 

Po P, 

indications now are h^ = 0.9964 and ^ = 1.0035. 

■Tl i 2 

27 



418 THE UNIVERSAL METHOD 

P P P 

its results are p" = 0.9880 and ~ = 0.9916, whence j- — 

0.9797, wrongly indicating a fall of 2.03 per cent. Yet ap- 
plied to the mass-quantities reduced as before at every period, it 

P. 

is consistent in always yielding p"' = 1.00 ; for its two serial 

1 

.P., P 

indications now are reciprocals, being p^ = 0.9895 and p^ = 

^1 ^2 

1.0106. Also in the several examples above given (in § 2), 
which are variations upon one theme, in all those in which the 
total sums at each period are the same. If rofessor Lehr's method 
rightly indicates constancy ; but in the one in which the total 
sum spent at the second period is larger than at the first, Pro- 

P., 

fessor Lehr's method yields p^ = 1.1081, which is 10.81 per 

cent, too high. ^ Thus in order to get the result known to be 
true, Professor Lehr's method requires a preceding reduction 
(except in rare cases when the proper conditions happen to ex- 
ist) before it is to be applied — a preliminary labor the need of 
which its author never contemplated. 

§ 4. When money has not the same exchange- value at the 
second period as at the first, one of the conditions necessary for 
proving the correctness of Scrope's mtsthod used in the way under 
discussion (and also of Professor Lehr's method) is absent^ — since 
constancy of money's exchange-value resulted from the two 
conditions posited in the above demonstration and was indispen- 
sable to it. Hence even if we make this preliminary reduction, 
— or if it is superfluous, the state of things already being as de- 
sired, — we cannot use the above demonstration to justify a belief 
that in all cases Scrope's method applied to the arithmetic 
means of the mass-quantities will give the correct answer. A 
single negative instance is sufficient to dispel any such belief. 
We may suppose the following state of things : 



I 100 A @ 1.00 100 B@1.00 
II 50 A @ 2.00 133iB@ .75 



100 for [A] 100 for [B] — 200, 
100 for [A] 100 for [ B] — 200. 



* Cf. this result with that above given for Scrope's method applied to the arith- 
metic means. From these and some other examples it would seem as if this form 
of Scrope\^ method and Lehr's method ^rr nearly equally on opposite sides of the 
truth. 



OTHER MKTHODS KXAMIXED 419 

As the weighting is even and only two classes are dealt with, 
this is a case in which all the three superior methods agree. 
Their common result is most easily obtained hv the geometric 

method. This is " = \/2 x / = ^^3 = l.'2'247, indicating a 

rise by 22.47 per cent. But Scrope's method a])pli('d to the 
arithmetic means of these mass-(|uantities is 

P,^75 x2+11G| X i ^ 237i _ 

Pj 75 X 1 + 116f X 1 191 f '^^ ' 

which, indicating a rise by 23.^*1 ])er cent., is slightly above 
the truth.' 

For further comparison another more irregular example may 



be subjoined : 






I 40 A @ 1.25 70B(<^1.50 
II 50 A fS) 1.80 100 B@ 1.30 


50 for [A] 105 for [B] - 
90 for [A] 130 for [B]- 


- 155, 
-220. 



Here the geometric method (in which the weighting is 1 for [A] 
and 1.7416 for [B]) indicates a rise of the price-level by 4.30 
per cent., the method with double weighting a rise by 4.31 
per cent., and Scrope's emended method a rise by 4.33 per cent. 
— all three almost exactly alike. On the arithmetic means of 
the full mass-quantities, Scrope's method indicates a rise by 4.21 
per cent.; and on the arithmetic means of the mass-quantities 
reduced so that their total money- values are the same at both 
periods (the numbers of A's at the second period being reduced 
to 35.225, and that of B's to 70.45) Scrope's method is still 
slightly wrong, as it indicates a rise by 4.3() per cent.'' 

What is needed more than a reduction of the total money- 
values to the same figure is the reduction to the same figure of 
the total exchange-values. Thus the first of these exam})les 
would need to be altered into the following : 

■'' Applied here to the smaller mass-quantities at either period, Sci-ope's method 
indicates a rise by only IGn per cent. — a considerable error. — Lehr's method here 
indicates a rise by 21.05 per cent. 

''Applied here to the smaller of the original mass-quantities at either period 
Scrope's method indicates a rise by 5.16 per cent.; and applied to the smaller of 
the reduced mass-quantities, a rise by 3.60 per cent. — Lehr's method applied to 
the original mass-quantities indicates a rise by 4.46 per cent.; and applied to the 
reduced mass-quantities, a rise by 4. .S3 per cent. 



420 THE UNIVEESAL METHOD 

I 100 A @ 1.00 100 B @ 1.00 
II 61.235 A @ 2.00 163.293 B @, .75 

100 for [A] 100 for [B]— 200, 
122.47 for [A] 122.47 for [B] — 244.94. 

And now Scrope's method on the arithmetic means of these 
mass-quantities is : 

P, 80.617 X 2 + 131.646 x | 

Pj ~ 80.617 X 1 4- 131.646 X 1 ^'^^^^^ 

which is exactly right.'^ This exact agreement, however, is due 
to some peculiarity in this example ; for we do not find it al- 
ways. Thus in the second example, after reducing the total 
sum at the second period to 161.68 (the number of A's then be- 
ing 36.745 and that of B's 73.49), on the arithmetic means of 
the mass-quantities Scrope's method indicates a rise by 4.27 per 
cent.* But we have reason to believe that this method, so used, 
would generally give a more nearly correct result than when 
applied to the arithmetic means either of the full mass-quantities 
as they happen to be at each period or to the mass-quantities 
reduced merely to the same total money-values. Still, to em- 
ploy Scrope's method in this way requires that we should first 
know the constancy or variation of the exchange- value of money. 
Hence Scrope's method, so used, is unsuitable for finding 
the constancy or variation of the exchange-value of money, 
since this use of it presupposes the knowledge we are search- 
ing for. 

Nor can the method of approach be employed with any great 
security. Thus in the first of the above examples, after the 
measurement by Scrope's method applied to the arithmetic mean 
of the mass-quantities as they actually are, which has been seen 
to indicate a rise by 23.91 per cent., if we provisionally assume 
that prices have so risen, we might work over the schema into 
the following : 

■^ Here also the right result is given both by Scrope's method applied to the 
smaller of these mass-quantities at either period, and by Lehr's method. 

* This calculation has been made on the assumption that the indication given 
by the method with double weighting, at 1.0431, was the right one. — On these 
reduced mass-quantities Lehr's method now indicates a rise by 4.34 per cent. 



OTHER METHODS EXAMINED 421 

I 100 XOi}, 1.00 100 BfS), 1.00 
II (;i.9r) A r'i) 2.00 165.21.3 Br^i) .75 

1100 for [A] 100 for [B] — 200, 
I 123.91 for [A] 123.91 for [B] —247.82; 

and now, applied to the arithmetic means of these mass-quan- 
tities, Scrope's method indicates a rise by 20.04 per cent. 
Again assuming this to indicate the price variation, we might 
repeat the operation thus : 

I 100 A @ 1.00 100 B@1.00 
II 60.02 A @ 2.00 160.053 B@ .75 

100 for [A] 100 for [B] — 200, 
120.04 for [A] 120.04 for [B] —240.08; 

wlience the indication is of a rise by 22.61 per cent. And 
again : 

I 100 A @ 1.00 100 B @ 1.00 
11 61.305 A @ 2.00 163.48 B @ .75 

100 for [A] 100 for [B]— 200, 
122.61 for [A] 122.61 for [B] —245.22; 

whence the indication is that prices in general have risen by 
22.46 per cent. And if we repeated the operation once more, 
we should practically get the same result again, or perhaps a 
closer approach to 22.47, according to the extent to which we 
carry the decimals. Therefore, having got practically the same 
result from two successive calculations, we should have no 
reason to go further, and should adopt this result, — which we 
already know to be the right one. But here again there is 
something peculiar in this example, that permits the getting 
of the right result ; for the right result is not always got in 
this way. Thus in the second of the above examples, for 
which the first employment of Scrope's method applied to the 
arithmetic means of the full actual mass-quantities gave us a 
rise by 4.21 per cent., if we assume this to be right and reduce 
the total money-values accordingly (that of the second period to 
161.5255), and work out as before, the indication is of a rise by 
4.34 per cent.; and again, assuming this, and reducing the total 
money- values (of the second period to 161.727), and working 
out as before, the indication is of a rise by 4.53 per cent.; and 



422 THE UXIVERSAL METHOD 

next, after reducing the total money-values (of the second 
period to 162.0215), the indication is of a rise by 4.35 per 
cent. This fourth answer hardly diifers from the second, 
wherefore the next would hardly diifer from the third, and 
pursuing these operations indefinitely we might perhaps reach 
some unvarying result somewhere between 4.34 and 4.53, but 
never could we get the right result, which lies outside these 
limits. Yet even if the right result could be reached in this 
way, the method would be unserviceable, as no more laborious 
method can be imagined. 

§ 5. The peculiarity in the first example which caused it 
twice to yield the happy result, while the results in the other 
example were always divergent, is, in all probability, the same 
peculiarity in it that caused the three superior methods exactly 
to agree. This is the fact that in it the two classes are equally 
important over both the periods. Thus, in general, when two 
classes equally important over both the periods are dealt with, 
the same result is given not only by the superior methods, but 
also by Scrope's method applied to the arithmetic means of the 
mass-quantities after these have been reduced so as to be pur- 
chased at each period with sums of money possessing the same 
exchange-value, and by Professor Lehr's method applied to the 
mass-quantities similarly reduced, and even by the former of 
these methods (and probably by the latter also) when applied to 
the mass-quantities reduced so as to be purchased simply by the 
same sums of money, if thus continued through the method ot 
approach. Here is additional confirmation that in such cases 
the common result so reached is the true result. But in other 
cases, or employed without such reductions or corrections, these 
two purely arithmetic averages diverge and scatter their results 
more than do the three at least partly geometric methods.^ 
Hence the latter (or at least t'ss^o of them) are probably nearer 
the truth than either of the purely arithmetic methods. 

Now it may be conceded that the classes with rising prices and 
the classes with falling prices may frequently, or in the long 
run generally, be about even. Therefore the purely arithmetic 

^ And as above noticed in Note 4 generally on opposite sides of the three. 



A GENERAL TEST' CASE 42:^ 

methods, if properly corrected through the method of approacli, 
would g'cuerally yield results almost as good as those given by 
the partly geometric methods. Yet, to repeat, the use of those 
methods with such correction is impracticable. 

But even without attempting such laborious correction, when, 
in practice, many classes are dealt with, and the variations both 
of prices and of mass-quantities are such moderate ones as gen- 
erally take place from year to year, containing many sources of 
compensation, and the total money- values of the classes not vary- 
ing much, nor the general exchange-value of money, then it is 
probable that Scrope's method a})plied to the arithmetic means 
will deviate but slightly from Scrope's method applied to the 
geometric means, and that Professor Lehr's method will deviate 
but slightly from the method with double weighting above ad- 
vocated. 

§ 6. For the })urpose of illustrating these inductions, and also 
of casting a parting glance at other methods, we may use a more 
complex suppositional exam])le than any used as yet. In order 
to magnify the errors, we may suppose very great variations 
both of prices and of mass-quantities ; and also, to give oppor- 
tunity for the working of compensatory errors, we may extend 
the comparisons over a short series of periods. I>et us, then, 
posit the following schema : 



I 6 A @ 2 


12 B @ 3 


10C@1 


3D@8 


6E@4 


1 F@6, 


II 4 A @ 3 


10 B@ 7 


6C@ 2 


4D@5 


6E@3 


2F@2, 


III 5 A@3 


13B@4 


8C@2J 


3D@4 


7E@3 


3F@2, 


IV 5 A @ 4 


12B@ 6 


9 C@2 


2D@6 


5E@5 


2F@4, 


V 7 A@2 


11 B@7 


7C@3 


5D@3 


8E@4 


3F@3, 


VI 6 A @ 2 


12 B Of) 3 


10C@ 1 


3D@8 


6E(5>4 


1 F @ 6. 



Here we have five periods with different states of thmgs, be- 
tween whicK are four different sets of variations. In addition 
it is supposed that at a sixth period everything returns to exactly 
the same state as at the first. The object in making the last 
supposition will appear presently. 

Measuring every set of price variations separately by the three 
methods above advocated, Ave obtain the following percentages 
(the positive indicating rises and the negative falls) — by the 
geometric method : 



424 THE UXIVERSAL METHOD 

+ 31.79, - 23.02, + 40.05, - 6.18, - 24.01; (1) 
by the method with double weighting : 

+ 31.72, - 22.92, + 39.98, - 6.36, - 23.64; (2) 
by Scrope's emended method : 

+ 31.11, - 23.00, + 39.93, - 6.35, - 23.39. (3) 

These yield the following index-numbers, the columns repre- 
senting the lines similarly numbered : 





(1) 


(2) 


(3) 


I 


100 


100 


100 


II 


131.79 


131.72 


131.11 


III . 


101.45 


101.53 


100.95 


IV 


142.10 


142.02 


141.27 


V 


133.29 


132.98 


132.30 


VI 


101.29 


101.54 


101.35 



Now by the arrangement of the prices and of the mass-quan- 
tities at the sixth period we know that the index-number for 
the sixth period ought to be 100. Our methods, then, err at 
the sixth period by between 1^ and IJ per cent, above the 
truth.^*^ If they contained a single sort of error which gradu- 
ally accumulated at every advance, we might correct the pre- 
ceding figure by reducing it by four fifths of 1;^, 1^ or IJ per 
cent., according to the method that is being corrected, and the 
next preceding by reducing it by three fifths, and so on. But 
this we cannot do, because we know that the errors in these 

i^Eather curiously it is with weighting according to the arithmetic means of 
the full money-values that the geometric average of the price variations gives, at 
the sixth period, in this example, the best result. Its indications then are of the 
following percentages : 

+ 33.17, --22.91, 4-40.12, -6.33, -25.16; 
forming these index-numbers : 

100, 133.17, 102.60, 143.80, 134.69, 100.80. 
AVith weighting according to Gauss's means (see Note 3 at the end of Sect. III. 
above) the geometric average gives these indications: 

-h 32.48, -22.98, -F 40.13, -6.26, -24.60; 
100, 132.48, 102.04, 143.00, 134.04, 101.07. 
With weighting according to the arithmetico-geometrico-harmonic means (there 
also suggested ) , the results are nearly the same, being 

4-32.47, -22.99, 4-40.09, -6.24, -24.57; • 
100, 132.47, 102.03, 142.93, 134.01, 101.08. 



A GENERAL TEST ('ASE 425 

methods arc not cinnulativo, but alternating. The fact that 
these methods show errors above the truth by certain amounts at 
the sixth period, gives us no hint as to the errors at any of the 
preceding periods. For instance, we cannot know whether the 
indications given for the fifth period are above or below the 
truth. In obtaining the sixth index-number from the fifth, we 
have measured a variation of the general price-level from the 
fifth U) the sixth period, which is just the inverse of the whole 
variation from the first to the fifth period. Hence to test the 
fifth period by the sixth period would be the same as to test the 
serial method of reaching the result for the fifth period by the 
direct method of comparing the fifth period immediately with 
the first — which latter method, as already said, has no better 
claim upon our approbation than the former. 

Nor would such a test be better than a test by comparing the 
fifth period directly with the second. In fact, if we use such 
direct comparisons at all, we ought to use all possible ones. 
\^'e ought to compare not only every period with the first, but 
every period with the second, every period with the third, and 
so on. In the following table the index-numbers resulting from 
such comparisons are stated, the method with double weighting 
alone being employed : 





Every period 

compared with 

the first. 


Every period 

compared with 

the second. 


Every period 

compared with 

the third. 


Every period 

compared with 

the fourtli. 


Every period 

compared with 

the fifth. 


1 


100. 


75.91 


94.42 


66.48 


76.36 


11 


131.72 


100. 


129.73 


91.83 


98.88 


III 


105.90 


77.08 


100. 


71.43 


79.97 


IV 


150.40 


108.89 


139.98 


100. 


106.79 


y 


130.95 


101.13 


125.04 


93.64 


100. 


VI 


100. 


75.91 


94.42 


66.48 


76.36 



For better comparison these figures may be re-arranged as follows, 
in the same order. 

I 100. 100. 100. 100. 100. 

II 131.72 131.72 137.39 138.13 129.49 

III 105.90 101.53 105.90 107.44 104.72 

IV 150.40 143.44 148.25 150.40 139.85 
V 130.95 133.22 132.42 140.84 130.95 

VI 100. 100. 100. 100. 100. 

As no one of the.se series is preferable to another, we may 



426 THE UNIVERSAL METHOD 

average them, — doing so on the first table, doubling the first 
column in order to represent also the comparisons with the sixth 
period, and using the arithmetic average for want of a better. 
The series thus obtained is as follows : 

100, 133.26, 105.28, 145.46, 132.84, 100. 

Yet such a series cannot be accepted as authoritative. 

Nor are better results obtained by using these methods ad- 
apted to a whole epoch — say of the five different periods. The 
geometric method, with weighting according to the geometric 
averages of the full money- values of the classes over all the five 
periods, yield these index-numbers : 

100, 140.0, 109.5, 155.7, 141.9, 100.^' 

The method with double weighting applied to the geometric 
averages of the prices of the mass-units over all the five periods, 
these : 

100, 131.01, 106.02, 149.49, 135.12, 100.^' 

Scrope's method applied to the geometric averages of the mass- 
quantities over all the five periods, these : 

100, 130.12, 104.38, 141.04, 133.93, 100.^^ 

None of these is satisfactory. It is plain that if another period 
were included in the epoch at the beginning or at the end, all 
the indications would be shifted. Thus, for example, with the 
sixth period added to the epoch, the method Mdth double weight- 
ing applied to the geometric averages of the prices over all the 
six periods yields these index-numbers : 

100, 134.49, 105.07, 149.76, 132.15, 100.^* 

No correction of the figures first given, therefore, seems 
to be possible. All that we can do is to take them as approxi- 

^^ With weighting according to the arithmetic averages : 

100, 140.3, 109.6, 155.8, 142.3, 100. 
^2 Applied to the arithmetic averages : 

100, 136.77, 105.89, 146.57, 135.12, 100. 
^ ^ Applied to the arithmetic averages : 

100, 128.76, 100.17, 140.80, 132.43, 100. 
^* Applied to the arithmetic averages : 

100, 140.18, 104.95, 149.13, 141.28, 100. 



A CENERAL TE8T CASE 427 

niately represent in j»; the true variation in the general level of 
prices. 

§ 7. Let ns now turn to the indications yielded by other 
methods applied to this same schema, beginning with those 
which make most pretension to accuracy. 

Professor Lelir's method gives the following percentages of 
the separate price variations : 

+ 31. ()9, - 22.8(5, + 39.84, - 5.70, - 28.09; 
which yield these index-numbers : ^ 

100, 131.69, 101.58, 142.05, 133.96, 96.33. 

These are very close to those given by the three methods 
similarly used in serial form, except only in the last indication.''^ 
Scrope's method presents many varieties. We have seen that 
two ways of drawing a mean between the mass-quantities at the 
two periods have previously been recognized. One of these is 
to apply it to the arithmetic means of the mass-quantities. 
This gives the following percentages : 

_|. 30.80, -23.00, -h 40.04, -6.96, -22.43. (1) 

Another is to make the calculations once on the mass-quantities 

''The method al)Ove suggested in Note 9 in Sect. II. gives these percentages : 
+ 31.71, -23.06, +40.2(3, -6.58, -24.09, 
which yield index-numbers as follows : 

100, 131.71, 101.34, 142.14, 132.78, 100.79, 
which also are very close. On the other hand, the method suggested in Appendix 
C, VI. § 3 as a possible form of Nicholson's method gives the following percent- 
ages: 

+ 33.01, -12.95, +55.70, -9.10, -2.41, 

which yield these index-numbers : 

100, 133.91, 116.57, 181.50, 164.98, 160.99, 
— the worst of all. The method (with geometric average of the inverted [mass- 
quantity variations) there also (in Note 4) suggested as a variant upon the last, 
would give these percentages : 

-^23.83, -22.17, +48.00, -19.57, -12.86, 
forming these index-numbers : 

100, 123.83, 96.38, 142.64, 114.72, 100, 
which are better, but still not good. It may be noticed that it is impossible to 
apply Drobisch's method to this example without further specifying the weights 
(or capacities) of the mass-quantities. Various suppositions with regard to these 
would make Drobisch's method yield various results, but would have no influence 
upon the results given by any of the other methods. 



428 THE UNIVERSAL METHOD 

of the earlier periods aud again ou the mass-quantities of the 
later periods, and to average the results. Thus Scrope's method 
on the mass-quantities of the earlier period in every comparison 
(the same as the arithmetic average of the price variations — 
Young's method — with the full weighting of the earlier period) 
gives these percentages : 

-h 40.18, - 22.79, -I- 42.06, - 1.29, - 14.29. (2) 

And Scrope's method on the mass-quantities of the later period 
in every comparison (the same as the harmonic average of the 
price variations with the full weighting of the later period), 
these : 

+ 21.43, -23.17, +37.78, -11.58, -30.87. (3) 

The arithmetic means between these percentages are the follow- 
ing percentages : 

+ 30.80, - 22.98, + 39.92, - 6.44, - 22.58. (4) 

In stringing these results out in a series of index-numbers 
there is a split into two ways of forming the mean series. The 
one is to form the index-numbers on the (arithmetic) mean per- 
centages, as given in line (4). The other is to draw the (arith- 
metic) means of the index-numbers in the two series formed on 
lines (2) and (3). The series so formed is added in column 



I 


(1) 

100. 


(2) 
100. 


(3) 
100. 


(4) 
100. 


(5) 
100. 


II 


130.80 


140.18 


121.43 


130.80 


130.80 


III 


101.72 


108.22 


93.29 


100.74 


100.75 


IV 


142.45 


153.74 


128.53 


140.96 


141.13 


V 


132.53 


151.76 


113.65 


131.88 


132.70 


VI 


102.80 


130.07 


78.57 


102.10 


104.82. 



The results in the average methods (in columns 1, 4 and 5) are, 
like those of Professor Lehr's method, very close to those given 
by the three more theoretically correct methods, showing that 
in practice all these methods are likely to -give results very 
nearly alike. The figures in columns (2) and (3) are, singly, 
extravagant and absurd. But there is order in their extrava- 
gance ; for the nearness of their means to the more truthful re- 



A (iENERAL TEST CASE 429 

suits .shows that they straddle the true course, tlie one varying 
on the one side about as the other does on the other.""' This is 
as we might expect from reasoning a priori ; for we have seen 
that the one has no more reason for it than the other, and there- 
fore the one has no more reason against it than the other. Be- 
tween the various ways of drawing the mean there is little ciioice, 
exee})t that tiie first is the most convenient.'" 

As for Young's method, when it is employed with what we 
have found to he pro})er weighting, though not proper for it, 
this gives most absurd results. Thus, employed with weighting 
according to the geometric means of the full money- values at 
both periods, its indications for the price variations are 

_f.o5.6(), -19.7(j, + 4;J.()2, +1.44, - O.ns, 

forming tliese index-numbers : 

100, lo5.(j(j, 124.90, 179.38, 181.97, 1X0.91. 

Or Avith weighting according tx) the arithmetic means, the per- 
centages are 

+ ;)7.12, -19.30, +43.67, -0.57, -1.8S, 

' ^ Of course the proper mean to draw here is really the geometric. The per- 
centages of the geometric means l)etween the variations whose percentages are 
given in lines (2) and (3) are 

+ 30.47, -22.92, +39.90, -6.58, -23.02 
which yield these index-numbers : 

lUO, 130.47, 100.48, 140.57, 131.23, 101.10. 
1' We might also use the geometric average of the price variations in the way 
the arithmetic has just been used — twice applying it to the same price variations, 
once with the weighting of the earlier, and once with the weighting of the later 
period, and then drawing the geometric mean between the two .results. This 
yields the following percentages — of the price variations on the weighting of the 
earlier periods : ' 

+ 16.94, -26.01, +38.11, -7.25, -33.03; 

of the price variations on the weighting of the later periods : 

+ 48.12, -19.49, +41.80, -5.47, -11.58; 
of the geometric mean of these variations : 

+ 31.61, -22.82, +39.94, -6.36, -23.05; 
and the following index-numbers on the last percentages : 

100, 131.61, 101.58, 142.15, 133.11, 102.42. 
But these are no better than those directly obtained by the geometric average 
with the geometric means of the weights of both periods. 



430 THE UNIVERSAL METHOD 

and the index-numbers 

100, 157.12, 126.72, 182.05, 181.02, 177.61. 
Even with the weighting of the first period alone in every com- 
parison, — weighting which is proper for the arithmetic average 
of the price variations when the mass-quantities are constant, — 
this method still errs largely, although it is now much better 
than with mean weighting. The figures have already been 
given — under Scrope's method, with which, in one form, this 
method, so weighted, is identical. 

Furthermore there is the method of using the geometric 
average of the jarice variations with the weights that are the 
smaller at either of the two periods compared, and the variety of 
Scrope's method which applies it to the smaller of the mass- 
quantities at either of the periods compared. The former gives 
these percentages and index-numbers : 

-\- 26.30, - 23.24, -^ 39.33, - 1.60, - 19.72, 
100, 126.30, 96.95, 135.08, 132.92, 106.71; 

and the latter these : 

+ 31.63, -22.90, +39.09, -3.45, -28.09, 

100, 131.63, 101.16, 141.16,. 136.29, 97.92. 
These figures, although sometimes nearly right, are erratic, 
and untrustworthy. Nor is anything trustworthy reached by 
taking an average between them. 

§ 8. Lastly it remains to see what plan should be adopted 
when we are reviewing the course of the exchange-value of 
money in times past, for which complete and accurate data are 
not obtainable. To attempt to use the complex methods above 
recommended as the best would be pedantic ; for the figures 
posited in our example, especially of the mass-quantities, are now 
to be taken as only approximately correct. We must still use 
uneven weighting : but it can only be rough vmeven weight- 
ing. There are two separate and distinct systems ready for our 
adoption. 

We can either (1) draw some average of the total money- 
values of the classes during an epoch of years, and with weighting 



A GENEHAL TF.ST CASH 4:^1 

SO determined (iinploy the geometric average of the [)rice vari- 
ations ; or (2) draw some average of the nuiss-(|uantities of the 
classes during the epoch, and apply to them Scro})e's method. 
In each case, in getting the average w eigiits or mass-quantities, 
we might as well, for greater convenience, employ the arithmetic 
average, because in these complex instances there is little differ- 
ence in the results whether we use the arithmetic or the geometric 
average, and neither is exactly true. But if ice use an average 
of the total money-values for our weighting, we must use the geo- 
metric average of the price variations ; and if we use an average 
of the mass-quantities, we must use Scrope^s method. There must 
be no intermingling of these operations. 

Both these methods so used, we have seen, universally satisfy 
Professor Westergaard's test. Hence it is indifferent whether 
we compare the periods successively, or compare each succeeding 
period with the same original base (whether at the beginning, 
end, or in the middle, of the series). The calculations whereby 
the rough weighting is obtained should be renewed from time to 
time — say in every decade. This will require the starting of 
new chains, from new bases. But the results may, if desired, 
all be strung out in one series. 

On our schema these two methods have already been worked 
out with the exact weighting, and applied to the exact averages 
of the mass-quantities, over all the five periods.**^ The results 
so obtained by Scrope's method, in this particular instance, are 
better than those obtained by the geometric method ; and in 
other cases we may be pretty sure they will be as good, if not 
better. Scrope's method is, then, recommended by its greater 
convenience. Over the five periods in question the exact arith- 
metic averages of the mass-quantities are: 5| A, 11| B, H C, 
3| D, 6| E, 2^ F. If in their places we used only rough 
averages, as e. g., in the same order, 5, 11, 8, 3, 6, 2, applied 
to these Scrope's method would give the following series of 
index-numbers : 

100, 127.77, 100.112, 140.74, 132.40, 100, 

^* Above in Notes 12 and 13. 



432 THE UNIVERSAL METHOD 

which, it will be noticed, are still fairly accurate. But there 
are occasions when, havmg a general idea of the relative im- 
portance of the classes, we may more conveniently use the geo- 
metric average of the price variations weighted accordingly. 

That the distinctive elements in these methods should not be 
mixed together, in violation of our above canon, may be shown 
by the results obtained from the application of Young's method 
to this schema. The following index-numbers are obtained by 
arithmetically averaging the price variations at the successive 
periods, all compared with the first, in every comparison with 
the same weighting, which is the arithmetic average of the total 
money-values over the five periods : ^^ 

100, 163:66, 123.93, 165.87, 172.16, 100, 

— the extravagance of which runs to absurdity. 

By the same schema may be tested a few more methods that 
have been put into practice. Jevons's method, also employed 
by Walras and Simon, using the geometric average of the price 
variations with even weighting, gives these index-numbers : 

100, 101.50, 92.47, 130.76, 104.63, 100, 

indicating these percentages of variation between the periods : 

^_1.50, -8.90, -f- 41.40, -19.98, -4.41. 

And Carli's method, as extended by Evelyn, and as employed, 
for instance, in the Economist, drawing the arithmetic average 
of the price variations with even weighting, gives the index- 
numbers : 

100, 125.70, 115.28, 144.44, 136.80, 100, 

indicating these percentages of variation between the periods : 

-f- 25.70, - 8.29, + 25.30, - 5.29, - 26.90. 

Thus in this example, as Jevons himself claimed, his method 
gives lower results than the ordinary arithmetic method. But 
its results are more aberrant even than those given by the latter 
method. , 

19 The same weighting as above used with the geometric average of the price 
variations in Note 11. 



Tin-: Missi.\<; I'ltoi'osi'noxs supplied 433 

The correction of tlie latter method offered bv Mr. Piilirravc, 
arithmetically averat>ini!; the price variations of the later periods 
compared with the first, in every comparison with the weij^htino- 
of the later period, apj)ears, in this example, worse than what it 
Avonld im))rove upon ; for its index-numbers are 

100, 171.07, l;n.41, 170.05, 174.58, 100, 

which are comparable only with those above given by Young's 
method, of which this method itself is a variety. In fact, 
Young's method, in every form, has been found to be bad.^" 

It must be remembered, however, that in actual practice the 
deviations of these methods would be much smaller than they 
are in this exann)le. 

V. 

§ 1. As a resume of the little which, after all these investiga- 
tions, we have learned with absolute exactness and positiveness, 
we can now supply, in part, two propositions that have been 
lacking hitherto. 

The one of these, whi(!li was missed in Chapter VII., is in re- 
gard to compensatory price variations. Because of the perfect 
accuracy of the geometric average with even weighting, or the 
geometric mean, of the price variations in all cases dealing with 
two classes between whose full Aveights at each of the two 
periods compared the geometric means are equal, we have this : 
Of two <'l<(x.scs equally important over both the periods compared 
compensalorii price variations are simple geometric variations, 
that is, variations from unity to the opposite geometric extremes 
around unity, so that the geometric mean between them is unity, 
indicative of constancy (Proposition XLY^II.). In other words, 
if the price of [A] rises by p per cent, to 1 -\- p times its former 
l)rice, in order that the exchange-value of money remain con- 

p 1 
stant, the price of fB] must fall by -^ per cent, to :; of 

its former price ; and reversely a fall of [A] by p' per cent, to 

-" With such weighting so used the harmonic average of the price variations 
yields better, but still unsatisfactory, results, its index-numbers being 
100, 147.02, 99.21, ir30.48, 11(3.65, 100. 
28 



434 THE UXIYERSAL METHOD 

1 — p' of its former price would be compensated by a rise of 

v' 1 . . 
[B] bv —- — , ])er cent, to , times its former price, — pro- 
vided the accompanving states, or variations, of the mass-quan- 
tities are such as to make these classes equally important over 
both the periods (which variations, if they be of the mass-quan- 
tities together purchased at each period with the same total sum 
of money, will likewise be found to be simple geometric vari- 
ations). This proposition is true in a world with only two 
classes of commodities beside money, if the classes so vary in 
price (and in mass-quantities). It is true also in a world with 
any number of classes of commodities, referring to two of them, 
provided the others do not vary in price, or vary in such wise 
that they compensate for one another (in pairs, or otherwise) 
and do not cause an alteration in the exchange-value of money 
(according to Proposition XXXI.). These others being such, 
their numbers will, as we have seen (in Proposition XIX.), 
cause the general exchange- value of money to fall less through 
the single rise in price of the one class than it \vould fall with- 
out them ; but they will cause the general exchange-value of 
money to rise equally less through the smgle fall in price of the 
other class. And the influence of these price variations singly 
to deflect the general exchange-value of money being reduced in 
the same proportion (according to Proposition XXV.), the com- 
pensation is the same, no matter how numerous the other classes 
may be, or how large. Or, if the other classes have varied so 
as to cause a variation in the general exchange- value of money, 
the influence of these two classes is such as it would be if their 
prices had remained constant. This last, however, belongs to 
the next Proposition. 

But we cannot extend this Proposition so as to say that 
among three equally important classes (over both the periods 
compared) the rise of [A] by p per cent, to 1 -|- p times its 
former price is compensated by a fall of [B] and [C] each by 



a percentage carrying its price down to ^ of its former 

" N 1 -|-p 

price, that is, to the figure between which twice repeated and the 



TILE MTSSINCi I'ROl'OSITHKNS SUI'I'LI KD 435 

other the geometric aiieracfe, with even \veij2;htinj>' foi- the three 
classes, is unity, indicative of constancy ; for this is not true, — 
and yet it is very near tiie truth. Nor can w(! extend it to any 
more complex cases, or to cases with uneven weighting. Yet 
in all these, with the amount of unevenness in weighting and of 
variation in prices likely to be met with in practice, the state- 
ment that comj>ensatory ])ri(;e variations must be such that the 
geometric average of them, with projx'i- weighting, should be 
unity, is likely to be nearly correct. The |)ecnliarity of the 
geometric average must always be borne in mind, that, while 
the geometric mean ])roper, between only two ecpially important 
classes, and therefore wdth even weighting, in all cases where it 
can rightly be used, yields the true result, the geometric (irerac/e, 
between more than two classes, or between une({ually important 
classes, wherever uneven weighting has to be em])loyed, yields 
a result deviating above or below the truth according to what 
seem to be fixed laws, but only slightly erring in ordinary cases. 
Wherever in practice, among a large number of classes, some 
large and some small, some rising and some falling in price, the 
geometric average with its proper weighting indicates constancy, 
we have good reason to believe that the true answer would show 
at most but a trifling variation. 

The other Proposition, which was missed in Chapter VI., is 
more comprehensive, — in fact, it embraces the preceding. It is 
this : When of firo classes equalli/ important over both the periods 
compared, the price of the one remains stationart/ while that of tit e 
other varies, or the prices of both rari/, the injiaence of these jtrice 
variations upon the general exchange-value of money is the same as 
it would be if both the classes varied, alike in the geometric mean 
between their .actual price variations (Proposition XLVIII.). 
This is the only definite answer that can universally be given to 
Jevous's problem, to which a threefold answer was given in the 
last Chapter, on the condition there posited that the mass-quanti- 
ties are constant over both the periods. The present Proposition 
is ajiplicable whenever the geometric mean between the full 
weights of one class at the two periods is the same as the geo- 
metric mean between the full weights of one other class, what- 



436 THE UNIVERSAL METHOD 

ever be the variations of the mass-quantities, provided, of 
course, they be such as with the price variations to permit of 
equality between the means. It directly covers, therefore, all 
the cases in Chapter X. in which equal sums are supposed to be 
constantly spent on two classes at both periods. It covers also 
those cases in Chapter XI. in which the mass-quantities are 
supposed to be constant, provided the sums spent at each period 
have this property of equality between their geometric means. 

But agam this Proposition cannot be extended to permit us 
to say that the influence of the price variations of any t^vo 
classes, or of any number of classes, is the same as it would be 
if they all changed alike in the geometric average between their 
actual price variations weighted in the manner always pre- 
scribed. And yet in most cases their influence would be very 
nearly such. Again what is true of the geometric mean is not 
exactly true of the geometric average. 

In the absence of an absolutely true method of measuring 
variations of general exchange-value, it is not convenient to im- 
prove upon the preceding general statements by attempting to 
formulate propositions in accordance with either of the other two 
methods of measurement which we found reason to believe to be 
slightly better than the geometric method. Yet by means of 
the formulae for these methods the influence of the price vari- 
ations of any numbers of classes, or, given certain price varia- 
tions, the compensatory price variations of any numbers of other 
classes, all the other conditions being stated, is easily found 
with still closer approximation to the truth. 

§ 2. If, however, it ever happens, or if we suppose, that the 
mass-quantities are constant through both the periods, thus cov- 
ering all the cases in Chapter XI., it is easy to summarize the 
principles there discovered and to state the lacking Propositions, 
confined to these special cases, in twofold forms with perfect 
definiteness and universality. The first is : M^ith constant mass- 
quantities, compensatory price variations are arithmetic variations 
with the loeighting of the first period, or harmonic variations with 
the weighting of the second period (Proposition XLIX.). The 
second : With constant mass-quantities, the influence upon the 



THE MISSING PROPOSITIONS SUPPLIED 437 

(/encrdJ exchange-va/ ac of vioiiey of ant/ two or more price varia- 
tions /.s the same as if the prices all varied in the arithmetic 
averaf/e with the weighting of the first period, or in the harmonic 
average with the weighting of the second period (Proposition L.). 
Tlie third forms in which these Propositions can also be stated 
have already been inclnded in the preceding Propositions. The 
last two Propositions have little chance of application. But 
what we know about Scrope's emended method shows that 
similar, but over-complex, Propositions could be framed, applied 
to the geometric means (and in most cases even to the arith- 
metic means) of the mass-quantities of every class at the two 
periods, that would very closely approximate to the truth. 



CHAPTEE XIII. 

THE DOCTEINE OF THE CONSERVATION OF EXCHANGE- 
VALUE, AND THE MEASUREMENT OF EXCHANGE- 
VALUE IN ALL THINGS. 



§ 1. In Chapter II. our Proposition XXXIX. was worded : 
" All things collectively, provided they be the same (or similar 
and in equal quantities) at all the periods compared, are con- 
stant in exchange-value." Examination of this principle was 
deferred. We are now prepared to examine it. 

This proposition is not new. In one form or another it has 
often been intimated. Originally, and most frequently, it has 
been stated in the form that it is impossible for all things to rise 
or fall in value. It was so stated by Senior as early as 1836, 
and by J. S. Mill in 1848, after the latter of whom it has con- 
stantly been repeated.^ Sometimes little more has been meant 
than the obvious assertion that neither all commodities together 
can rise in exchange- value nor all commodities together can fall 
in exchange-value. But usually there has also been an impli- 

^ Senior, Political economy, p. 21 (originally in the Encyclopsedia Metropoli- 
tana, 1836 : also in Arrivabene's translation of Senior's writings, Paris 1836, p. 
104); Mill, op. cit., Vol. I., pp. 540, 588 ; Levasseur, B. IS, pp. 138, 158 ; Courcelle 
Seneuil, op. cit., Vol. I., pp. 259, 272; J. Bascom, Political economy, Andover 
1860, p. 224; Fawcett, op. cit., p. 312; Bolles (quoting Faweett), op. cit., p. 55: 
Cairues, op. cit., p. 12; Macleod, Theory of banking, Vol. I., p. 70, Theory of 
credit. Vol. I., p. 176; A. L. Perry, Introduction to political economy, 1881, p. 
65 ; Mannequin, Question monetaire, 1881, p. 7 ; Jourdain, Cours d'economie 
politique, 1882, p. 435 ; Hansard, B. 67, p. 1 ; E. T. Ely, Introduction to political 
economy, 1889, p. 179 ; S. N. Patten, Theoi'y of dynamic economics, 1892, pp. 64-65, 
and Cost and expense, 1893, p. 57 (of "objective values"); Denis, B. 125, p. 171 ," 
Fonda, B. 127, p. 6; Laughlin, op. cit., pp. 147-148; Bourguin, B. 132, p. 25; 
Parsons, B. 136, pp. 114, 115 (of "internal exchange- values," offering a dem- 
onstration which is mathematically incorrect). 

438 



THE DOrmiNE DKSCRIHKI) 489 

cation of couiitci-balancing/ with the meaning- that all conimod- 
ities tojicther can neither rise nor fall in exc^hangc-valiie — in 
other words, they are constant in exchange- value.' In this 
sense it is true only if the commodities are the same (or similar, 
in equal mass-quantities) at all the [)eriods compared. This 
proviso has been noticed by CJairnes, who added that " the 
aggregate amount of values " may increase or decrease with an 
increase or decrease of comiuodities.^ 

The need of this proviso may be made a})})arent by analogy. 
If we have a universe consisting of three atoms endowed with 
equal attractive force, the attractive force of each atom is one 
third of the whole attractive force in the universe. If then a 
fourth atom is created in this universe, endowed with attractive 
force equal to that of each of the others, although the force in 
each of the atoms now sinks to one fourth of the whole, it is one 
fourth of a whole larger by one third, and ^ x (o -(- 1) = ^^ X •^, 
so that each of the three atoms continues to have the same force 
as before, and as there is one more such atom, the attractive 
force in the universe has increased. Similarly if we have an 
economic world consisting of three articles, exchanging equally 
for one another, so that they are endowed with equal purchasing- 
power or exchange-value, the purchasing power possessed by 
any one of these is one third of the whole purchasing power in 
this world. Then if another article is produced in this world, 

- Souietime.s erroneously. Thus Ely : " Let us suppose that to-day two bushels 
of wheat exchange for three of oats, and to-morrow for four bushels of oats. We 
may say that wheat has risen in value, but it is obvious that exactly in the same 
proportion oats have fallen in value," loc. cit. Wheat has risen 33i per cent, in 
exchange-value in oats ; oats have fallen 25 per cent, in exchange-value in wheat. 
These proportions are not the same. There is geometric equality of proportion ; 
but this should be specified. — A mistake like this is responsible for Parson's 
attempted demonstration. 

* This inference has been directly drawn by Fonda (without the proviso), B. 
127, p. 18. — In this sense, but ambiguously about "value," the proposition has 
been combated by A. Clement, De I'iufluence cxercee par la hausseou la baisedes 
valeurs siir la richesse genVrale, .Journal des Economistes, .July 1S54, pp. 8-10> 
because he perceived that use- values can rise or fall together ; and by Pollard, 
op. cit., Ch. IX., because it is not true of cost-value, in which sense alone he pre- 
fers to use the term "value." Cf. also Ricardo above quoted in Chapt. I. Sect. 
IV. Note 8. 

* Op. cit., pp. 12-13. The proviso has also been incorporated in the statement 
by Mannequin, loc. cit. 



440 THE CONSERVATION OF EXCHANGE-VALUE 

which will exchange for any one of the others, so that it is en- 
dowed with pnrchasing power equal to theirs, although the 
purchasing power of each of the articles is now one fourth of 
the whole, yet as it is one fourth of a whole larger by one third, 
as in the former instance, therefore the purchasing power of each 
article continues the same as before (cf. Propositions XLII. and 
XLIII.), and so the total amount of purchasing power, or ex- 
change-value, in this world has increased. 

§ 2. There is, however, a form in which this principle may 
be presented which seems to make it true ev'en under any 
changes in the mass-quantities.^ This form is, in fact, in accord- 
ance with the way in which the principle was originally hit upon. 
Let us conceive an economic world containing two articles, A 
and B, equally valuable at first. If A comes to rise in ex- 
change-value compared with B, we know that B must fall in 
exchange- value in A . Therefore there cannot be a rise both of 
A and of B in exchange- value each in the other, or of both in both 
together ; nor contrariwise a fall of both together. Now if A 
becomes twice as valuable as B, B becomes half as valuable as 
A, and from having A == B and B === A at the first period, Ave 
have at the second A === 2B and B -== JA. How are we to con- 
ceive of a constancy here of the exchange-values of A and B 
together? If we add together the equal exchange- values of 
A and B, each taken as a unit, at the first period, we get the 
sum 14-1 = 2; and if we add together the unequal exchange- 
values at the second period, to wit, 2 for A and ^ for B, we get 
2 -f- I = 2^, which is not the same as before. Or again, A has 
risen by 100 per cent, in B, and B has fallen by 50 per cent, in 
A, that is, A has varied by -|- 100 per cent, and B by — 50 
per cent., and if we add these we get -|- 50 per cent., which 
does not indicate constancy. The indication of constancy is to 
be got in another way. At the first period we may multiply 

5 This wider position cannot be argued for on the ground that the exchange- 
values of all things could change only relatively to something else, which is self- 
contradictory ; for they also could not remain constant, in this way, except in 
relation to something else, which is equally self-contradictory (cf. Chapt. III. 
Sect. II. §^ 4 and 9). What we are here dealing with is not the exchange-values 
of all things as a whole compared with something else, but a total of some sort 
of all tlie exchange-values of all things within a whole. 



THE DOfTKIXE DESCRIBED 441 

the exchaiige-valut' of cuch article as expressed in the other, and 
the product is unity, for 1x1 = 1; and at the second period 
the multiplication of the exchange-value of each article as ex- 
pressed in the other still yields unity as its product, for 
2 X 2 == 1- Yet if a third article had come into existence, say 
with the exchange-vahie of B, on multiplying together the ex- 
pressions for its exchange-value in A and A's in it, for its in B 
and B's in it, and, as before, for A's in B and B's in A, as fol- 
lows, 1x^x1x1x2x1, the product still is unity. 

These considerations lead to two propositions, which are 
curious, but of little utility. They are : The product of the 
ratios of all paHicidai' exchange-values is unity (Proposition 
LI.) ; and, The product of all variations of particular exchange- 
values is unity (Proposition LIL). To prove these propositions 
we may try them on some simple case, so far as possible in uni- 
versal form. Let us suppose that we have an economic world 
containing three classes, such that we have, at the first period? 
one mass-unit of [A] for every two mass-units of [B] and for 
every three mass-units of [C] , and that the mass-unit of [A] is 
worth 6^ times the mass-unit of [B] and Cj times the mass-unit 
of [C] . Then all the ratios of exchange-value between these 
things are expressed in the following list : 

A==6iB c 6iB =^ c^C=== c^C =c. c^C, 

B =^ IB =^^A-=^^C^^C=c=^^C, 
6j b^ b^ 6i 

Be. IB^cAa c^C-^Jc^^Jc, 
6i 6i b^ b, 

C^^^B .^^ B c=iA^c^ IC^-IC, 

Ci c, Cj 

C^AiB c^^B c^^ A=c^ IC^ IC, 

Ci Cj c^ 

C <:- ^-' B c. ^ B =^ ^ A == IC o IC. 

Cj Cj Cj 

And now in multiplying all these ratios together we find we 



442 THE CONSERVATION OF EXCHANGE-VALUES 

have tAvo 6j to multiply by two r j three Cj to multiply by 

1 . c ,6 

three — , six ^ to multiply by six — , and eight 1 to multiply 

together, the product of all which is unity. Then at the 
second period if we suppose A to be worth b^ times B and 
c^ times C, and if we suppose the same or any other num- 
ber of mass-units of these classes, or even any other classes 
with any other ratios of exchange-value, treating them in the 
same way, we should get the same result. , But when we mul- 
tiply together the variations, it is plam that we have variations 
only of the same numbers of mass-units in each of the classes. 
Let us then suppose we have at both periods those already 

supposed. The variation of A relatively to each B is j^ and of 

each B relatively to A r^ , of A relatively to each C - and of 

c be 

each C relatively to A - , and of each B relatively to each C j~^ 

be. 
and of each C relatively to each B ~-^ , while the variation of 

"1^2 
each B to each B and of each C to each C is 1. Thus the full 

list is : 

Variations of A — 



62 62 C2 C2 C2 
b' b"' t' c' t 



B-1 



B-1 



6j 6jC2 b^a.^ b^c^ 

62 ' 626^ ' 62^1 ' ^2^1 ' 

^1 ^\^-l \^2 ^1^2 

b/ b.-^G^' b^c^' bf 



21 



6,02' 61C2' C2' ' 

61C2' 6^03' C2' ' 

Q_^_£i fei 'h 1 1 

^^2' 61^2' C2' ' 



THE DOCTRINE DESCRIBED 448 

And here, too, it is [)erceive(l that we have only even numbers 
of reciprocals, so that the multiplication of all these variations 
by one another <:;ives the product, unity. And the same wouhl 
be the result with any numbers of things, with any particular 
variations. 

These Propositions, then, are of little importance theoretical 
or practical ; for as the products of all the ratios at every period, 
and of all the variations between any periods, are always unity 
— always Tthe same, — we gain nothing by comparing them." 

But there is more in the Proposition placed at the head of 
this Chapter, something which renders it truly a doctrine of the 
conservation of exchange- value (of a certain kind), under tlic 
proviso that the classes and their mass-quantities are the same 
at both periods, — or that we confine our attention to those 
classes, and to those portions of them, that are the same (or 
similar) at both the periods compared. In order to understand 
this we must find the method of measuring, under the proviso 
stated, the variation of the exchange-value of every other thing 
under consideration, as well as that of money, and not only of 
their exchange- values in all other things, but also of their ex- 



hans'e-values in all thinirs. 



II. 

§ 1. Under the condition that the mass-quantities are the 
same at both periods (or that we are dealing only with the mass- 
quantities that are common to both periods) we know that 
Scrope's method, applied to the constant mass-quantities, is the 
proper method for measuring the variation of the price-level, 
and the inverse of this is the proper measurement of the vari- 
ation of the exchange-value of money in all other things. Let 
us now examine, under its universal aspect, the simplest hypoth- 
esis possible, namely the familiar one of a world with money an<l 
two commodities. We have generally stated it in the following 

schema 

I .r A @ «! y B @ l\, 
II .1- A @ a, 2/ B @ (i^. 

® Cf. a similar useless position in Chapt. V. Sect. I. § 5. 



444 MEASUREMENT OF EXCHANGE-VALUE IN ALL THINGS 

This is sufficient when we are treating of the exchange-value of 
money in all other things ; but it is incomplete when we are 
treating of the exchange-value of money in all things. Then 
we must include a statement of money itself, the quantity of 
which must, in accordance with the hypothesis, be the same at 
both periods, and the price of its money-unit is, of course, at 
both periods a unit. Hence, with the volume of money repre- 
sented by V, the schema becomes 

I x A @ a^ y E@ Pi I' M @ 1, 
II .r A @ a^ 2/ B @ /32 v M @ 1. 

Scrope's formula for the variation of the exchange-value of 
money in all other things is 

this being the same as 

in which «/' = xa.^ -f- ■ii[-i^ , and this again the same as 
ifo2 1 / a, 62 



Mo, 






that is, as the arithmetic average of the exchange-value vari- 
ations with the weighting of the second period. We have 
already learned, m Chapter V. Section YIL, to turn such a 
formula for exchange-value in all other things into the formula 
for exchange-value in all things. This is, returning to the 
second of the above formulae. 






in which n.^" = xa,^ -f- yj3.^ -\- vl. This reduces to 

which is Scrope's formula for the variation of the exchange- 
value of money in all things (including itself). 



WITH CONSTANT MASSES 445 

Besides this we want Serope's forniiilie fV»r the variation 
of the exchan^e-vahie in all things both of [A] and of [B]. 
These we may obtain by simj)ly imitating what we have done 
for money. In obtaining the formula for money we first stated 
tlie conditions in terms of money, the money-unit being taken 
as the unit of excliange-vahie at eacli j)eriod, and the exchange- 
vahies of the othei' mass-units being stated as they relate to the 
exchange-value of this money-unit at each j)oint. Therefore, 
seeking the formula for [A], we must first of all state the rela- 
tions of the exchange-value of the other mass-unit and of the 
money-unit to the exchange-value of the mass-unit of [A], 
taken as unit at each period. The schema then is 

I X A (,(\ 1 V ^(w ^ V M (,<', \ 

n x A Oi\ 1 iiB(a\'-^ vM(a),-. 
«., a. 

As we have here modelled everything u])on the previous schema 
of money, merely changing the imits, we may merely copy the 
formula for the variation of the exchange-value of money in all 
things ; and so we have 

/^i 1 

4 •'■ + ^a+V 

X + y + " — 

«2 «2 

which reduces to 

Then for the case of [B] we have, first of all, the^schema : 
II / A @ -5^ y B (a^ 1 vM.Qt. -T-; 

Pi ' Pi 



whence the formula, 



446 THE CONSERVATION- OF EXCHANGE- VALUE 



«, 1 



-Pa2 Ih Hi 

/'2 /^2 

^ i^ii^'h + y/^i + t^ ) _ (^ . ^2 

P'i(a'«2 + .'//^2 + ^') -^1 ^4i* 

What is thus shown for money and two classes of commod- 
ities is plainly perceived to be extendible to money and any 
number of classes. With any number of classes, for any class, 
say [L] , whose prices are -^.j and X^, we shall have 

Lai ~ K Mar' 

We thus have a convenient method for converting the formula 
for the variation of the exchange-value of money in all things, 
under the given proviso, into a formula for the variation of any 
one class (that was included in the formula for money) in ex- 
change-value in all things. We only have to multiply the ex- 
pression for the variation of the exchange- value of money in 
all things by the price variation of the class in question.* 

§ 2. Having obtained these formulae, we may now discover 
the meaning of the Proposition at the opening of the Chapter. 
AVe have seen, by a particular example, which is sufficient to 
prove a negative, that the expressions for the exchange-values 
of all things (or classes) in all other things do not sum up to the 
same figure at each period, even when the same things are in 
existence at both periods. We shall now find that, given this 
condition, the universal expressions for the exchange-values of 
all things, or classes, in all things do sum up to the same figure, 

^A similar result cannot be obtained with exchange-value in all other things ; 
for, while we have 

Mo-2 x a^+yji^ 

Mq] ~ xa. + yji^ ' 
we have 

^0 -2 _ a-ziyl'^i +^') , ^02 _ /JgC-TOj +v) 

Aoi^ a^{yl32+v) ^^ Boi^ (i^ixa^ + v)' 

and there is no such reduction. — All this is in agreement with Proposition 
XX XIII.; and it suggests a still broader Proposition. 



THE ])()(TI!1NK EXPLAIXKI) 447 

when treated in a special inann(M". This is to multiply the ex- 
pressions for the exchange-values at each period by the weights 
at the first period. These expressions at the first period are to 
be taken as units, so that the sum of these units nndtiplied by 
the weights of the first period is thus expressed, xa^ -j- i/^i^ -\- v. 
And at the second period, (;omj)arcd with the first, the expres- 
sions for the exchange-value in oJ/ things of all the classes are 
the expressions above discovered for the variations of the ex- 
change-values of all the classes in all things. Hence what we 
want to prove is 

a., 3f^., ^ /9., Mao 3fa2 

^". ■ »; ^ M„ + •'«'■ ■ ft, ■ j/„; + " • K, = ""' + ■'"'' + " • 

The first half reduces, and we get 

(x«2 + ?//9, -f- V) - = .ra^ + j/,.:^, + v, 

which is evident when we restore the full ex])ression for -J* . 

Q. E. D. Thus at any tivo periods in which the mass-qaantitie^s 
are constant the sums of the exchanf/e-valiies in all fhinr/s of all 
the classes (or of all things) iimUiplied by the laeights at the first 
period are the same, so that comparison of them gives unity, indi- 
cating constancy (Pro})Osition LIIL). 

What 'is thus easily proved, may be varied in the following 
way. Of the three expressions 

Ma,' O.; Ma,' /9/l/«t' 

we know that at least one must be below, if one is above, unity, 
and reversely. We know this on general principles already 
reviewed." It also follows as a necessary consequence . from 
what has just been proved. But we may also gather it by sim- 
ple inspection of the expressions, when stated in full. Let us 
suppose that the exchange-value of money has risen and that 

1/., 

^rp = /• [r being > 1). The prices of [A] and [B] nuist now 

■'■'■Lai 

1 Propositions IX., XV., XVI., XXVIIL, XXIX. 



448 THE COXSERYATIOX OF EXCHANGE- VALUE 

either have both fallen uniforml}', or at least one of them must 
have fallen. If they have both fallen uniformly, then, in the 
full expression, v being a positive quantity that cannot be neg- 
lected, — and '-T^ must each be greater than r ; and consequently 

— and '^ must each be smaller than - . But r multiplied bv a 

figure smaller than - gives a result smaller than unity. Or if 

one of the prices has fallen less (or remained constant, or even 
has risen), then the other must have fallen by still more than 

- , and at least the expression for the variation of the exchange- 
value in all things of the class (and everything in it) having this 
price variation must be below unity. The same result will 
follow if we suppose the exchange- value of money to have fallen, 
or if we started with either of the other classes ; and also it will 
be obtained if we enlarge the supposition to include any number 
of classes. Always, if any one of the expressions is on one side 
of unity, at least one of them must be on the other side. (Where- 
fore if all but one are known to equal unity, that one also must 
equal unity, in obedience to Proposition XX,). 

This being so, it admits of demonstration that the percentage 
of the variation, or the sum of the percentages of the variations, 
of exchange-value in all things, above unity, multiplied by the 
weights at the first period, is equal to the sum of the percentages 
of the variations, or to the percentage of the variation, of ex- 
change-value in all things, below unity, likewise multiplied by 
the weights at the first period. We do not know, in the ex- 
ample before us, which of the above three expressions is larger, 
or which smaller, than unity. There are six possible permuta- 
tions, and we may assume one of them as a specimen. Let us 
assume that money has risen in exchange-value in all things, 
and that the two commodity classes have each fallen in ex- 
change-value in all things. Then the percentages of the rise 

of money (in hundredths) is -jp — 1, and the percentages of the 



riii; ixxTiiiM-; kxi'I>ai.\ki> 449 

falls of the commodities are 1 ■ , ''" «*ii<' ^ — ^ ' ir'"- ^'^e 

therefore Avish to prove that 

And thi.s is easily done because the etiuation to be proved re- 
duces, as before, to 

jt;,, 

and so a<j;;ain is evident Avhen we restore the full expression for 
the variation of money. And if we suppose any of the five 
other permutations, we shall always find a similar demonstra- 
tion. And what is thus proved of money and two classes, may 
be extended to money and any number of classes, or to cases 
without money ; for, although the demonstration becomes more 
complex with a greater numljer of classes, it may be carried out 
in the same manner. 

The meaning of what has just been proved may be expressed 
as follows : The same mass-quantities of all classes existing at both 
periods, the sum of the percentages of the variation of every class 
in exchange-value in all things, each multiplied by the weight of 
the class at the first period (the percentages of the rises being treated 
as positive (juantities, and the percentages of the falls as negative 
qncDititics), is zero. (Projiosition LIV.) This zero means that 
the common variation of all things together in exchange-value 
in all things is by zero per cent., Avhich is constancy. 

Here we have used percentages in the ordinary Avay, reckon- 
ing them in the starting points of the variations — at the first 
period. Hence it is only natural that importance should be as- 
signed to the variations according to the importance of the vari- 
ants at the first period. And conceiving the projjortions of the 
variations in this the usual way, we find that, when the mass- 
quantities are constant, there is what we have called arithmetic 
equcdity between all the proportions of the rising variations of 
exchange- values in all things and all the proportions of the fall- 
ing variations of exchange-values in all things, the weights of 
29 



450 



THE COXSERVATIOX OF EXCHANGE- VALUE 



the classes at the first period being assigned to their variations. 
This is precisely ^vhat we should expect in such cases, because 
of Scrope's formula holding here. For we have learnt that 
Scrope's formula is the same as the formula for the arithmetic 
average of variations with the weighting of the^j's^ period. 

We may also expect more, since we know more about Scrope's 
formula. We may suspect that there also is, in these cases, 
what we have called harmonic equality between all the propor- 
tions of the rising variations of exchange-values in all things 
and all the proportions of the falling variations of exchange- 
values in all thmgs, the weights assigned to the variations be- 
ing those of the second period. In other words, the sums of 
the percentages of all the rising and of all the falling variations 
harmonically reckoned, that is, reckoned in the finishing points, 
at the second period, multiplied by the weights at the second 
period, will be equal ; or, if added together as positives and neg- 
atives, will yield zero as the total sum. This may be quickly 
proved by taking the same permutation in the above example, 
and stating its percentages (in hundredths) harmonically. We 
wish, then, to prove that 



f J4 
3L 



- 1 



^a2 
ill., 



: XO..^ 



a. M„, 



J 



and can do so, because this reduces to 

Ma, 
Ma, 

and to 



\._^2 M_a, 



v\l 



= xa. 



L /3i Ma 

^ Ma 



/«! Mar ,\^ , (^ Ma, \ 



Ma 

xy±nn 



M, 



which is evident when we restore the full expression for ^r^ 

(the inverse of -^ \. Q. E. D. And again, as we may ex- 
pect, we find that, with two things, or classes, constant in mass, 
and equally important over both the periods together, there is 



THK DOCTinNK KXI'LAINKI) 40] 

r/roincti'lc eqiiali/t/ between the iiroportions of their variations in 
exchange-value in all things (viz., the two things or classes tliem- 
selves), — that is, the ])ercentages of tlieir variations in such ex- 
change-value, geometrically reckoned (reckoned in opposite di- 
rections) are e(iual. For, supposing these articles to be priced 
in ideal money, [A] rising and [B] falling, what we have to 
prove is 

// " + '^' •>■ 'J -I- .'/ 



I'l 



y „ + ^ 
2 



but this reduces to 



//V^l/^2 = ^^''h"-l ' 



whi(!h is the very condition supposed, so that the equation is 
true when this condition is satisfied. Q. E. D. We should 
find also, by trial, that in ordinary cases there is approximation 
to geometric equality, or equality between the sums of the geo- 
metric jiercentages of all the rising variations together and of all 
the falling variations, if these percentages are multiplied by the 
weights over both the periods. But these two positions, about 
liarmonic and geometric equality, are of little utility, especially 
the last ; and they do not merit being stated in formal })ropositions. 

Thus we have found in what sense the Proposition at the 
opening of the Chapter is true. It is true of general exchange- 
value m all things rightly measured, and with proper allowance 
for the sizes of the classes. It is not true of general exchange- 
value in all other things.'^ Therefore we should add to it the 
words " In all things." 

§ 3. In evidence of what has here been universally proved, 
and as suggestive of other relations, a particular example may 
be oifered. Suppose we have these states of things : 

^ Parsons tried to demonstrate it of this kind of general exchange-value. 
Hence the necessity of his failure. See Notes 1 and 2 in Section I. of this Cliapter. 



452 THE CONSERYATIOX OF EXCHANGE- VALUE 

I10A@1 5B@4 6C(aj3 4D@7 7E^2, 
II10A@1 5B@4 6C@6 4D@7 7E@,2. 

Here we have only one price variation, and this price variation 
shows that the class [C] has risen in exchange-value in all 
other things by 100 per cent, (reckoned in the ordinary way). 
Now if we measure the variation of money in exchange- value in 
all other things by means of Scrope's formula, we have 

P2 _ 10 + 20 -H 36 -h 28 -h 4 _ 108 ^ 

P, "" 10 -j- 20 -h 18 -f- 28 + 4 ~ 90 ~ ' 

indicating a rise of 20 per cent., which means that money has 
fallen in exchange-value in all other things by 16f per cent, 
(for -j^Q^^g = 0.8333 ). This means also that all the com- 
modities which have remained unchanged in exchange- value 
relatively to money have also fallen by 16f per cent, in ex- 
change-value in all those things in which money has fallen by 
that percentage (that is, in all the things which are other to 
money, namely in all commodities). Now let us remove money 
from consideration, — or let us suppose we have been dealing 
only with ideal money (money of accomit). As it has not en- 
tered into the calculation, its removal causes no change. But 
now all commodities become all things. And the classes, [A], 
[B] , [D] , [E] , which remained unchanged relatively to money 
and to one another, are still altered in exchange-value in all com- 
modities, that is, now, in all things, just as they were before. 
Thus these classes have fallen by 16f per cent, in exchange- 
value in all things. And the class [C] has still risen by 100 
per cent, in all other things ; but as it has risen by 100 per cent, 
above things that have fallen by 16f per cent, in exchange- 
value in all things, it has risen from 1 to 2 x 0.83|= 1.66f, or 
by 66f per cent, in exchange- value in all things (not counting 
money, but including itself). This may be shown again by sup- 
posing that at the second period all the prices were 16f per 
cent, lower than they were supposed actually to be. Then we 
should have 

P,_ 81 -H 16f 4- 30 -h 23i -Hlf ^ 90^ 

P 10 + 20 + 18 -h 28 -f 14 90 ' 



THE DOCTRINE EXPLAINED 45.3 

indicntiiio constancy. For here, money remaining stable, the 
prici'. and consequently the exchange-value in all things other 
to nidiiey, that is, in all commodities,* of each of the classes 
has fallen by 16 1 per cent., except the price of C, which has 
risen by 66f per cent., showing that its exchange-value in all 
commodities has risen by (30 1 per cent. The same results could 
also have been obtained directly, by making use of the method 
above ex])l()ited. For here we could immediately take A as 
the unit, omitting money altogether, and should have 

J[„,^ 10 + 20-H8-h28 -f 14_^ 90 _ .. , . 
A^^ 1 + 20 -f 8(> + 28 4-14 108 ' ^ ' 

and for the variations in exchange-value in all things of B, D, 
E, we should have this multi])lied by |^, ^, |, but for C we should 

have it multiplied by |, with answer y^ = l-66f, showing the 

rise of [C] in exchange- value in all things to be by 66§ per cent. 

Now OGf is four times 1(3 §. But the weights of the classes 
[A], [B], [D], [E] are four times larger than the weight of 
the class [C] at thej^r.s^ period. Thus four classes, four times 
larger tlian the class [C] at the first period, have together fallen 
4 X 16|=(3(3| per cent., arithmetically equalling the rise of 
[C] by 66f per cent., in exchange-value in all things (namely 
the five classes of commodities).'^ 

^ 4. In this form the Proposition may be of some utility. 
Thus if we suppose only two classes equally important at the 
first period, and the mass-units of each, A and B, are equally 
valuable at the first period ; if, the mass-quantities being con- 
stant, the exchange-value of A rises to double that of B, we see 

' Also in all things (iucluding money); for otherwise the method of conversion 
described in § 1 of this Section would be at fault. This is because money, here, 
is constant in general exchange-value of both kinds. It is plain that any number 
of classes could now be added with constant prices (or in the preceding case with 
prices rising 20 per cent.), without affecting any of the relations thus far used. 
See also Proposition XXXIY. — But this indication is not true of their exchange- 
values in all other things (other to them singly). See Proposition XXXIII. 

' Again, at the second period the four classes together ai'e twice as important as 
[C]. Reckoned harmonically, ('. e., in the finishing points, the percentage of 
their falls is 20 per cent., and the percentage of the rise of [C] is 40 per cent. — 
just twice as much. 



454 THE CONSERVATION OF EXCHANGE-VALUE 

that, whereas at the first period each class had half the total ex- 
change-value in the world, at the second period [A] has two 
thirds of it and [B] one third ; wherefore, although A has risen 
by 100 per cent, in exchange- value in the other thing, and B 
has fallen by 50 per cent, in exchange-value in the other thing 
— simple geometric variations, — the exchange- value of [A], and 
consequently of A, in all things has risen from J to f by 33 J 
per cent., and the exchange-value of [B] , and consequently of 
B, in all things has fallen from |^ to |^ by 33|^ per cent. — which 
are simple arithmetic variations.^ And again, in an example 
like the one just used, if, [AJ and four other classes being each 
equally important at the first period, A rises by 100 per cent, in 
all the others, the exchange-value of [A] (and of A) in all 
things rises from ^ to | by 66|- per cent., while the exchange- 
value of each of the others in all things falls from 1 to ^, or 
together from ^ to |-, by 16| per cent. 

We can sum up these relations in the following general state- 
ment : — When in cm economic world the mass-quantities ai'e con- 
stant, the variation of the exchange-value in all things of any class 
{and of any individual in it) is the variation of its shai'e of the 
constant total exchange-value or purchasing power in this world 
(Proposition LV.). 

And from this prmciple and the preceding examples we can 
form the following universal statements about the relationship 

^ Thus here, dealing with exchange-value in all things, we have the equality 
in the sums which we could not get in Sect. I. § 2 when dealing with exchange- 
value in all other things. Take another simple example. Suppose B is twice 
as valuable as A at the fii'st period, and at the second A twice as valuable as 
B, and that the numbers of A's and B's are equal and constant. Then [A] 
from having J of the total exchange- value in the world has come to have 'i of 
it, and [B] from having | has come to have J. Thus arithmetically reckoned, 
A has risen by 100 per cent., and B has fallen by 50 per cent., in exchange- 
value in all things; but the weight of [B] is twice that of [A] at the first 
period. But again, the weight of [A] is twice that of [B] at the second 
period ; and now, reckoned harmonically (in the ending points), A has risen by 
50 per cent, and B has fallen by 100 per cent., in exchange-value in all things. 
And again, the weights of [A] and [B] over both the periods together are equal ; 
and now, reckoned geometrically (from the opposite points), the percentages of 
the rise and fall in exchange-value in all things are equal ; for [A] has risen by 
50 per cent, reckoned at the finish while [B] has fallen by 50 per cent, reckoned 
at the start, or reckoned in the opposite directions [A] and [B] have risen and 
fallen each by 100 per cent. This illustrates g 2 of this Section. 



THE DOCTIUNE EXPLAINED 455 

between the two kinds of general exchange-value. The mass- 
(juautities remaining constant, if the weight of a class at the 

first period is the sum of the weights of all the other classes, 

and if this class rises by p j)cr cent, (in hundredths, reckoned in 
tlie ordinary way) in exchange-value in all otJier things, it rises 

from 7 to z in exchange- value in a(/ things, or by 

/• + 1 /■ -f 1 -f- p "^ ^ ' -^ 

rp 

per cent. ; while the others, collectively, fall from 



r-\-l-i-p 



r '>' . 
zr to z in exchange-value in all things, or by 

P 

per cent., and if they do not vary amongst themselves, 

they each fall by this percentage, or otherwise they fall by this 

)ercentage on the average ; so that the sum of the percentages 

f their falls equals the percentage of the rise of the one class. 

A.nd if this class falls by p' per cent, in exchange-value in all 

1 1 — »' 
dier things, it falls from -, j—, to j , in exchange- 

/■/>' 
vine in a// things, or bv — -^ .percent. ; while the others, 

■ J- _L 1 _ n' J 



+ 
+ r ^ r -\- 1 —p' 



r r , 

cdectivelv, rise from to - q j in exchange-value in 

• ' I 4- r r 4- \ —to' ^ 



P 
althings, or by ; — A. -7 per cent., and if they do not vary 

an)ngst themselves, they each rise by this percentage, or other- 
wii they rise by this percentage on the average ; so that the 
sui of the percentages of their rises equals the percentage of the 
falof the one class. 

'ere we have an answer for one of our early problems, posed 
in !!hapter II. Section IV. § 2, — but an answer applicable 
onl to the cases when the mass-quantities are the same at both 
pei>ds. We see that the larger are the other classes (the larger 
the), the smaller is their variation, in accordance with Proposi- 
tioiXIX., and the more nearly the variation of the one class 
in 'cchange-value in all things comes to its variation in 



456 THE COXSERVATIOX OF EXCHAXCiE-YALUE 

exchange-value iu all other things, according to Proposition 
XXIV. 

§ 5. We can, however, very much simplify the statement of 
the relationship between the two kinds of general exchange- 
value, by starting at the second period. In the first complex 
example above used (in § 3) another relationship stared us in 
the face, although we have not paid attention to it yet. This 
is that the exchange- value of [C] in all things (commodities) has 
varied two thirds as much as its exchange-value in all other 
things, — or its variation is one third less in the former than in 
the latter ; but also the weight of this class at the second period 
is one third of the weights of all the classes, — or the weights of 
all the other classes are two thirds of the weights of all the 
classes. This relationship is universal, as is proved by the fol- 
lowing equation expressing it in regard to money (on the sup-i 
position that money is rising in exchange-value), 

V -\-xa^-\ry[i^-\- 



« + a^«2 + yi^2 + a^«2 + 2//^2 + 



a^^-i + M + _ -^ ^ + a;«2 + M + 

a'S + 2//52 + 

which is easily seen to be true (or if we suppose money to b 
falling in exchange- value and reverse the terms in the first haj 
the same relationship is evident). Therefore ' 

The mass-quantities all remaining constant, the variation of aij 
class in exchange-value in all things relates to its variation n 
exchange-value in all other things as at the second period le 
weights of all the other classes relate to the weights of all the class 
(Proposition LVI.).^ 

" This is the Proposition referred to under Proposition XXV. It shows tie 
importance of the word "equal" in that Proposition. Here among consnt 
mass-quantities, as one class alone varies in price, its size varies, and so therere 
does the proportion between the variations of its two kinds of general exchaje- 
value. Thus as the class rises in price, its size increases, and therefore the ri'of 
its exchange-value in all things lags farther and farther behind the rise oits 
exchange-value in all ofJ2er things. An ordinary class rising slightlyits 
exchange-value in all things rises almost as much as its ex change- value ifill 
other things. But it is conceivable that the class could rise so high as to becae, 
at the second period, equal to all the other classes together. Then its exchsge- 
value in all things has risen only lialf as liigh as its exchange-value in all ier 
things. 



THE DOCTRINE EXPLAINED 457 

This Proposition applies to the variations in the two kinds of 
general exchange-valne of every individnal thing in the class in 
qnestion, when by its " exchange-valne in all other things " we 
mean its exchange-value in all things outside its own class.^ 
But if by the exchange-value of an individual thing " in all 
otlicr things " we mean absolutely its exchange- value in all other 
things (including all other individuals in its own class) beside 
itself, the Proposition must be modified. Its two variations 
will now relate to each other as the total exchange-values (or 
money -values) of all other things beside itself and the total ex- 
change-values (or money- values) of all things relate to each other 
at the second period. 

III. 

^ 1 . The mensuration of exchange- value in all things may be 
extended to all possible cases — with mass-quantities also vary- 
ing. It is only necessary to insert the article itself, whose ex- 
change-value in all things is being measured, among the things 
in which its exchange-value is being measured, and to invert 
the formulae above found to be the best for measuring the varia- 
tion of the general level of prices. Thus for money the uni- 
versal schema is 



I i\ M @ 1 .»i A @ «! (/i B @ /?! 
II tv M @ 1 Jo A @ a., y, B @ ii^ 



The formula for the method using double weighting is 
il/„, t\ + x^a^ -j- y^/?i -f- v.^ + x^a^a^ -f 2/2^/^/2 + 



3/„j I', -\- .x-2«- -I- .y./i -\- 1', + .rX«i«2 + yy?x?-i -f 

Now it is evident that this formula, in relation with the 
formula for the same method of measuring the variation or 
constancy of the exchange-value of money in all other things, 
may violate Propositions XXII., XXVI., XXXII. and 
XXXY. This is merely a continuation of the defect dis- 
covered in the preceding Cha})ter. But the other two superior 

■'* This is the sense in which the term has generally been used in these pages. — 
Of course, the mass-quantities being constant, the variation in exchange- value in 
all tilings is the same of the individual as of its class. 



458 MEASUREMENT OF EXCHANGE-VALUE IN ALL THINGS 

methods are free from this defect ; wherefore m dealmg with 
exchange- value in all thiugs the employment of one of them 
becomes imperative. And the one to be preferred is Scrope's 
emended method. The formula for this method is 

M^ ^ ^^1^2 + «X^'i^2 + ^yyiy2 + 

Again, reckoned in the mass-unit of [A], the schema is 

I?)iM@- XiA@l 2/iB@-^ , 

IIi;2M@i a-2A@l 2/2 B @ §' 

The formula is 

A - ^^1^2 + ^•'»r^2 + '^ ^3/1^2 + 

x/tYi'2 + -^^v^^ + - >^2/i3/2 + 

'^•2 "-2 

(>/v,v2 4- o.yx^x.^ -f- ^jy^i^/a -f- ) ,. 

- (^^'1^2 + «2^^-'^V'i*2 + /^2<M2 + ) 



And, reckoned in the mass-unit of [B], the schema is 

Ii)iM@-g- a-iA@^i 2/1 'B@l , 

IltJjM®-^ a-, A@~^ VoB@l 

/52 " Pi 

And the formula, 

/^2 /'2 

1 



1 ~ 9 ^ha 



\\rTH VARvrxf; masses 459 

And so on tlin)n<>;h all the classes. These results are tlie same 
as ill the partial eases, with constant mass-quantities. They 
would be the same if we used the formula for the geometric 
method, or the formula for the method with double weighting 
Just rejected, or any other formula whatsoever. Thus in general, 
W/ien tJic expression for the variation of the e.vchaiif/e-va/ue of 
iiionci/ in all thiiuji^ has been obtained, the expresi<io)i for the vari- 
ation of the exeham/e-ra/ue of ant/ other clam in all thinr/.s inai/ 
be obtained by nmltiplying the former expression by the expression 
for the price variation of this class (Proposition I^VII.).' 

§ 2. Here it is impossible to find any equality in the sums of 
the percentages of the variations, multiplied by the weights of 
the first period, or in any other regular manner. For if what 
was found to be true in § 2 of the preceding Section were assumed 
to be true here, with regard to three classes, we should have to 
be able to show that 

but on restoring the present value of ^ ''', we find that this is 

not necessarily true, — nor would it necessarily be so if we used 
either of the other methods for measuring variations in exchange- 
value. It may be true sometimes, but only accidentally. There- 
fore, as Cairnes said, the aggregate amount of purchasing power 
in an economic world may increase or decrease. 

Such increase or decrease can be measured in the following 
manner. Take from all the classes the mass-quantities that are 
common to both the periods compared. As these are collectively 
constant in exchange-value in all things (amongst themselves), 
their presence may be ignored. To be examined are the ex- 
cessive mass-quantities of some classes at the first period and of 
some classes at the second period. If the exchange-value of 
money has been found to be constant, all that is needed is to add 
up the money-values of the excessive mass-quantities at the first 
])eriod, and the money-values of the excessive mass-quantities 

1 As before, this Proposition cannot be stated of exchange-value in all other 
things. It expands Proposition XXXIII. 



460 MEASUREMENT OF EXC'HAXGE-YALUE IX ALL THIXGS 

at the secoud period ; and the balance between them will show 
whether there is gain or loss, and to what extent. But if the 
exchange-value of money has been found to have varied, the 
operation must be corrected by reducing the total sum of the 
money-values at the one or the other period — say at the second 
b}' multiplying the sum first obtained by the expression for the 
variation of the exchange-value of money — and then striking 
the balance. 

This procedure does not show the relative increase or decrease. 
After all, then, the completest method is simply to add up all 
the raoney-valuesof the classes at each period, and, after reducing 
the sum at the second period to a smn in money with constant 
exchange-value, to compare the two sums together. 

Here we must be careful as to which general exchange-value 
of money it is whose variation is to be corrected. If we are 
measuring merely the increase or decrease in the aggregate ex- 
change-value of conunodities, having no interest in the increase 
or decrease in the total exchange-value of the class money, we 
should use merely the variation of money in exchange-value in 
all other things (that is, in all things other to it, therefore in all 
commodities), at the same time leaving out of account the quan- 
tity of money. 

After such a measurement has been made, and the increase, 
decrease or constancy of the aggregate amount of exchange- 
value in all things in an economic world has been ascertained, 
it must not be supposed that any information has been acquired 
as to the increase, decrease or constancy of the aggregate amount 
of use-value or cost- value in that world ; and whether the 
aggregate amount of esteem-value is thereby determined is a 
debatable subject. The truth of this statement may be seen by 
considering the conditions when the mass-quantities are all ex- 
actly the same at two different periods, — when we know the 
aggregate amount of exchange-value in all things is constant. 
For that the aggregate amount of the cost-values of these con- 
stant mass-quantities — determined by the aggregate amount of 
labor their jjroductiou has cost — need not be the same at both 
periods, is evident. Equally evident is it that the aggregate 



WITH VA1!VIN(; MASSKS 4H1 

amount of their use-values may not he the same. For instance, 
if the two periods are summer and winter, the aggregate of the 
ti>tal utilities of all the classes might be very different at the twit 
])eri()ds. On the other hand, it would he a ditlicnlt (juestiou to 
decide whether in this latter case (on the supposition also that 
the numbers of the ])eople are constant) the aggregate amount 
of the final utilities of all the classes, and hence the aggregate 
amonnt of the esteem- values of all the constant goods, are con- 
stant or not. For if the goods were, say, mostly adapted to 
summer, a few of them would have no (actual) esteem- value in 
summer and many no (actual) esteem-value in winter, while 
many would have moderate esteem-value in summer and a few 
excessive esteem-value in winter. Whether these would exactly 
counterbalance or not, is out of the province of this work to ])ass 
an o])inion. 

It is pro])er, how'ever, here to state that the old and much 
debated question concerning the measurement of the iceaffh of a 
country is to be decided by saying that such measurement is 
the measurement of the increase, decrease or constancy of the 
aggregate amount of exchange- value (of all commodities in all 
connnodities) in that country relatively to the numbers of its 
inhabitants. 

§ .3, Both because the aggregate amount of the exchange- 
values of all things in all things is not constant, and because the 
measurement of general exchange-exchange is not known with 
absolute definiteness, when the mass-quantities as well as prices 
vary, it is impossible to construct such convenient statements 
for the comparison of the variations of the two kinds of general 
exchani^e-value as were above obtained for the cases when the 
mass-quantities are constant. But by means of the geometric 
average of the price variations, with its proper weighting, which 
method we know to be approximately accurate (except only in 
rare and extravagant cases), we may obtain almost equally con- 
venient approximately correct formulte also for these more usual 

and complex cases. Let the class [A] l)e - the size of all the 

other classes of commodities over both the periods together, and 



462 MEASUREMENT OF EXCHANGE-VALUE IN ALL THINGS 

let US suppose its price has risen by p per cent, (in hundredths), 
while the prices of all other commodities have remained un- 
changed. Then the exchange- value of money in all other things, 
i. e., in all commodities, has fallen approxhiiately from its first cou- 

, r + l/ \ r + l/ 1 

dition, represented as unity, to v :r by 1 — ->/ -^ 

? 1 ^ ' ^ 1 _j_ p -^ I 1 _|_ ^5 

per cent. Therefore every one of the other commodities, re- 
maining unchanged in relation to money, has fallen approxi- 
mately by this percentage m exchange- value in all other things 
beside money, that is, in all commodities (including itself). 
But the class [A] , while rising by p per cent, in exchange- value 

in all other things, has risen by p per cent, above 



of its former state, represented as unity, in exchange-value in 

all commodities (including itself, and excluding money), or to 

p . p 
r + i/ above its former state, by ,- + i/z = — 1 per cent., 

approximately. Reversely if [A] has fallen by p' per cent, in 
price, and therefore in exchange- value in all other things, money 
has risen approximately in exchange-value in all things from 



1 to t/ , by ,/ , — 1 per cent. ; and the ex- 

y I —J) ^ i—p 

change-value of every one of the other commodities has fallen 
approximately by this percentage in exchange-value in all com- 
modities (including itself). But the class [A] has fallen by p' 

per cent, below ,/ ^ of its former state in exchange- 

p 
value in all commodities, or to r^ .. , =: below its former state 

by 1 — ; — per cent., approximately. Or if we desire 

' / 1 - P' 
to make similar measurements with money itself included in the 
standard, supposmg its weight over both the periods to be s 
times that of [A], we merely have to add this to the indices of 
the roots in the preceding expressions. The larger r is, it is 
plain the smaller will be the opposite variation of money and 



WITH vAitYixc; masses 46'> 

the other commodities in exchangc^-vahie in all tliiii<>;s, and the 
nearer will the rise of [A] in exchange- value in all things be 
to its rise in exchantre- value in all other things, according to 
Propositions XIX. and XXIV., already referred to in similar 
cases above examined. 

Again, if all prices vary equally by + p or by — j) per cent, 
(rising in the former case and falling in the latter), or if they so 
vary on the average (geometrically measured), their variation 
may be comprehensively represented as a variation from 1 to 
1 dzjO. Then, while the exchange-value of money in all other 

thiup's has in verselv varied from 1 to , either bv ^ or 

^ ' 1 ±:p ' I -\-p 

by ^ pel- cent., its exchange-value has varied approximately 

- 1 -p 

from 1 to ' (in which /• represents the number of times 

{l±p)r,-.' i 

all the commodities are larger in total exchange-value over both 
the periods together than money). Therefore a commodity 
which has varied in price by dzp per cent, (the same as the 
average of all the price variations) has varied from 1 to 1 ±p 
in money that has varied to the one or the other of the above ex- 
pressions according as the one or the other of its general exchange- 
values is considered. Thus in all things other to money (/. e., 
m alt commodities, including itself) this connnodity has varied 

from 1 to =1 or to = 1 , that is, it has remained 

1+7' l-p 

constant, — and so, of course, also in exchange-value in all other 
commodities ;• but not so in exchange- value in all other things 
(other to itself, or to its own class, including money), or in all 
things (including money). In all things it has varied from 1 to 

\~ '( - or to , '• , or from 1 to (ldz»), + i, while 

{l+p)r + . (l_p),...' ^ 

in all otJwr things it has varied from 1 to (1 ±/>)^ — approxi- 
mately. 

§ 4. In thus measuring the exchange-value of money in all 
things, or of anything else in all thmgs, including money, we 
must know the quantity of money that has been employed at 



464 MEASUEEMEXT OF EXCHAXGE-YALUE IX ALL THING^S 

each period. Here is a problem of a peculiar nature, never yet 
discussed. The mere comparison of the quantity of money in 
a country with the quantities of commodities in the market at 
any one time is not satisfactory, as money is permanently in the 
market, while other goods merely flow through the market. 
Nor is there any propriety in the comparison of the quantity of 
money in the country with the quantity of goods (reckoned in 
money- values) bought and sold during each of the periods com- 
pared, because the lengths arbitrarily chosen for these periods 
will affect the quantities of the goods but not the quantity of 
money. Nor again between the quantities of exchanges effected 
with money and the quantity effected without money, because 
this comparison is affected by banking expedients, and banking 
expedients, although they make, certain kinds of money less 
needed and more scarce, yet do not lessen the importance of 
money as a class — or if the quantity of money so decreased be 
counted for less, the quantity of the substitute, material or uu- 
material, provided by the banking exjjedients, ought to be 
counted, which is still more difficult to estimate. The follow- 
ing suggests itself as a possible solution of the question. Money, 
^vhen used as a mere medium of exchange, is desired only for 
purchasing other things, hence not in itself, and so is incom- 
parable with commodities. But money is desired in itself when 
it is wanted for paying debts or the interest on debts. Thus 
the quantity of debts falling due within a certain period (but 
only those contracted in an anterior period) along with the 
quantity of interest due on these and on unexpired debts, seems to 
be a quantity comparable with the quantities of commodities con- 
sumed within the period. Yet there is defect also here. One 
of the reasons for wishing money to be stable in exchange-value 
is because of this very debt-paying quality it possesses. But 
the larger the amount of debt in a country, and the larger the 
amount of weighting consequently assigned to the class money, 
the smaller will be the calculated variations of money's exchange- 
value in all things, whenever a variation takes place ; so that the 
greater the quantity of debt and consequently the greater the 
nicety desired in the calculation, the duller becomes the calculation. 



WITH VAI{VIN(; MASSES M)^) 

Perhaps, however, after all, we do not want a standard con- 
sisting of all things, including money, but only a standard of 
all commodities. This is the conuuodity-standard proper. 
Money is something ,sul (/cncris, which we commonly contrast 
Avith commodities. It is used as a measure of the exchange- 
value of commodities. Hence we want it to be stable in ex- 
change-value in all commodities. To be sure, if it be stable in 
this exchange-value, it is also stable in exchange-value in all 
things, including itself; wherefore no room is left for dispute in 
regard to the supreme desideratum. But when it varies in gen- 
eral exchange-value, and therefore differently in exchange-value 
in all things and iii exchange-value in all commodities, the lat- 
ter variation being slightly greater, the commodity-standard not 
only is the more convenient because of the greater and conse- 
quently more apparent variation of money in it, but also it seems 
to be tlie standard in reference to which any correction in the 
])ayment of debts ought to be made. Still, this is a fine ques- 
tion, which will become of practical importance not till society 
at large shall have grown much more sensitive to variations in 
the exchange-value of money than it is at present. For the 
present, then, we may assume the conuuodity-standard merely 
for its convenience. 

§ 5. Adopting this standard, we see that a variation in the 
price of any commodity exactly represents the variation of this 
commodity in exchange-value in all commodities only if money 
has remained stable in excliange- value in all other (and in all) 
things, or, which is the same thing, only if the general level of 
prices has remained the same. Yet people almost always re- 
gard a variation of the price of anything as an indication of its 
variation simply in exchange-value in general, that is, they treat 
money as if it were stable in general exchange-value — as if it 
were a good practical individual standard. But when it is per- 
ceived that money has not remained stable in exchange-value, 
there is need of correcting this false inference from the mere 
price variation of any article. The correction can be easily 
made if the variation of the general level of prices (inversely 
showing the variation of the exchange-value of money in all 
30 



46(i MEASUREMENT OF EXCHANGE-VALUE IX ALL THINGS 

commodities) has alreadv been measured. Thus if the general 
level of prices is found to have fallen by 10 per cent., which 
fall means that .90 money-unit is now worth what 1.00 money- 
unit formerly was worth, it is evident that a commodity Avhose 
price has also fallen by 10 per cent., so that, instead of bemg 
held at 1.00, a certain quantity of it is sold for .90, has re- 
mained constant in exchange-value (in the commodity-standard). 
And a commodity that has fallen 5 per cent, in price, so that 
Avhat of it before cost 1.00 is now got for .95, has really risen 
in such exchange- value from .90 to .95, which is a rise by 5.55 
per cent.; while a commodity that has remained constant in 
price, has really risen in this exchange- value from .90 to 1.00, 
which is a rise by 11.11 per cent. On the other hand a com- 
moditv that has fallen "20 per cent, m price has really fallen in 
this exchange- value from .90 to .80, or by only 11.11 per cent. 

Now as one of the uses we make of actual prices is to indicate 
the variations of commodities simply in exchange-value, and as 
this indication is false except in one given case, it may not be 
out of the way to call the corrected indication by the term true- 
price. Of course the actual prices are always true prices ; for 
they are always true in the only indication which they properly 
make, namely of the variation of the commodity in its particular 
exchange- value in money. By " true-prices " is meant some- 
thing which is true, not necessarily in the first and proper in- 
dication made by actual prices, but in the secondary indication 
to which prices are habitually put. 

This term, then, being permitted, we can most conveniently 
find true-prices by making use of the principle above enunciated 
in Proposition LVII. That Proposition, to be sure, refers only 
to the variation of the exchange-value of money m all things, 
and shows only the method of obtaining therefrom the variation 
of any commodity in exchange-value in all things. But when 
we know the variation of the exchange-value of money in all 
other things, which are all commodities, then by multiplying 
this by the price variation of any commodity, we obtain the 
commodity's variation in the same standard of all things other 
to money, namely all commodities. Thus the expression for 



WITH VAUVIN(; MASSHS 4<37 

the variation of money in exchange- value in all commodities, 
according to Scrope's nietliod, heing 

j/^o ^ o.yx^x., + ,'-iy !/,!/., + ry-,^2 + 

^^01 a.,\/x X., + ,iV.'/,.'/., + yy-r-s + 



the expression, according to the same method of measurement, 
for the variation of A in exchange-vahie in all commodities (in- 

A 

eluding itself), represented by j^ , would be 

^ac-l '^i "l 

■^aci / ,12/ , 1 1 / , 

«^ (sA'*^2 + ^yll^}h + /'y~r2 + ) 

and so on through all the list of commodities. Usuall}', how- 
ever, the first measurement would be in the form of obtaining 
the variation of the general level of prices. But this is merely 
the inverse of the variation of the exchange- value of money in 
all other things (/. e., in all commodities). Hence, the operation 
of measuring the variation of the general level of prices having 
already been performed, the formula for finding the variati(Mi of 
any class, say [A], in exchange- vahie in all commodities is 

Thus, in general : The variation of the exchange- value of aaij 
commodity in the commodifif-damlard ii^ found hi/ multiplying the 
inverse of the expression for the variation of the general level of 
prices by the expression for its oion price variation (Proposition 
LYIII.). 

Or, if preferred, percentages alone may be dealt with. L(!t 
the percentage (always in hundredths) of the already measured 
variation of the general level of prices be represented by =b tt, 



468 GEXEEAL EXCHAXGE- VALUE 

according as it is a rise or a fall (if there be constancy^ - = 0) ; 
and let ± p similarly represent the percentage of the known va- 
riation of the price of the commodity in question. Then, the 
desired percentage of the variation of the true-price being repre- 
sented by pac , the formula is 

i±p-a^') 



1±7Z 

and this variation is a rise if the result be positive, a fall if the 
result be negative, and it is constancy if the result be 0. Or if 
the symbols be used to represent flill percentages (in uitegers), 
the formula is 

lOOzh/j -(100±-) 



I lOOzhn -(UX 

A. = 100 1 ^^^- 



It must always be borne m mind that when the variation of 
a commodity is so measured, the measurement is of its variation 
iu exchange-value, not in all things, nor in all other things, nor 
even in all other commodities, but in all commocUtie^ (including 
itself).- Of course, if the commodity be found to be constant in 
exchange-value in all commodities, and money also has been 
found to be constant iu exchange-value iu all other things, then 
the commodity is known to be constant iu all the other kinds of 
general exchange-value. Otherwise, to find its variation in any 
of the other kinds of general exchange-value would require a 
separate and special measurement similar to that by which the 
variation of monev was measured. 



IV. 

§ 1. AVhen engaged iu measuring the constancy or variation 
of the exchange- value of anythiug, it is essential that we should 
be clear as to the standard we are using ; or else there is danger 

- All this is said on the supposition that the commodity in question is one of 
the things whose prices were taken into account in measuring the variation of the 
general level of prices. If it be one of the things necessarily left out of that 
calculation, its variation in exchange-vahie determined as above is its variation in 
exchange-value in all other commodities (by "all" being meant all the other 
commodities that are included in the standard, it being impossible to include 
absolutely all). 



NEED OF DISTIXfrUISHING ITS KINDS 469 

of our falling into inconsistencies. This risk, and this need, may 
be illustrated by the following example. 
Suppose at the first period 

A = B = C = D c Ec^F = , 

and at the second 

A c fB = iC = D^=E»3=F=^ , 

the only changes being in relation to B and C And supj)ose 
the classes [B] and [C] are equally important over the two 
periods together. Then we know that the exchange-value of A 
has not varied. But at the second period 

B == 21 C = li A c 11 D r 11 E == 11 F o^ 

If B had risen so as to become equivalent to 1|^ of every other 
thing, we should say it had risen by 50 per cent, in general ex- 
change-value (of some sort) ; and as it has risen by this amount 
in all the other things beside [A] and by more than this amount 
in [A], it has evidently risen in general exchange- value (of 
some sort) by slightly more than 50 per cent. Also at the 
second period 

• C c AB. 2A.|D^=|E = |Fc^ ; 



and for a similar reason we perceive that C has fallen in general 
exchange- value (of same sort) by slightly more than 33 1- per 
cent. Now then we might be led into the following argument. 
While, at the second period, A is exchanged for | B, it is ex- 
changed for 33^ per cent, less of an article which has risen 
slightly more than 50 per cent. Therefore it has risen somewhat 
in general exchange- value ; which is contrary to our first conclu- 
sion. And while, at the second period, A is exchanged for 1^ C, 
it is exchanged for 50 per cent, more of an article which has 
fallen slightly more than 33^ per cent. Therefore it has fallen 
somewhat m general exchange-value ; which is contrary both to 
the first conclusion and to the immediately preceding. These 
two opposite conclusions cannot be reconciled by saying that they 
counterbalance each other ; for they would do so only if it were 
necessary for i\\e owners of [A] always to exchange it in equal 
portions for [B] and for [C] . But, according to the above 



470 GENERAL EXCHAXGE-YALUE 

reasoning, if anyone exchanged all his [A] for [B], he would 
at the second period, get more general exchange-valae than he 
got at the first ; and if another exchanged all his [A] for [C] , 
he would at the second period get less general exchange- value 
than he got at the first — which happenings are contradictory 
(because of Proposition VII.). Now the above reasoning would 
be perfect, and would lead us into a dilemma, or aporia, casting 
doubt over our whole subject, if the general exchange- values 
referred to in all the cases were the same. But they are not. 

§ 2. In the first case A is constant in general exchange- value 
of both kinds — exchange- value in all other things and exchange- 
value in all things. In the second case B rises by more than 
50 per cent, in a general exchange-value which is exchange- 
value in all the other things beside it, including C, but excluding 
itself. In the third case C falls by more than 33J per cent, in 
a general exchange-value which is exchange-value in all the 
other things beside it, including B, but excluding itself — and 
hence different from the exchange-value in other things in which 
B rose by more than 50 per cent. 

On the other hand, B in exchanging for 2 J C while exchanging 
for 1|^ of all the other things, exchanges for 2 J of an article 
fallen in exchange- value, while it exchanges for 1|^ of other 
articles with constant exchange-value. If then C has fallen in 
general exchange-value of some sort by exactly 33^ per cent., 

9 — 3 
2J C is equivalent to — - — = 1 1^ times the former C, and there- 
fore to 1|^ A, to 1|^ D, etc. And C in exchanging for only |- B 
while exchanging for f of the other articles, exchanges for -| 
of an article risen in exchange- value, while it exchanges for f of 
other articles with constant exchange-value. If then B has 
risen in general exchange- value of some sort by exactly 50 per 

4 + 2 
cent., I" B is equivalent to ^ — = f of the former B, and there- 
fore to f A, to f D, etc. Thus in these cases everything comes 
out right, provided that B and C have so varied in some general 
exchange- value, and that this is the same general exchange- 
value. 



nj:ki) of DisTiNfinsiiiNii its kinds 471 

Bat B and C have so varied, and in general exchange-value 
whi(^h is the same ; for they have both so varied in general ex- 
change-value in (iJI things (including themselves). For A has 
remained constant in exchange-value in all things, including B 
and C. Therefore, according to Proposition I^VII., when B 
has risen bv 50 j)er cent, in A, it has risen by 50 per cent, in 
exchange-value in (ill things, although it has risen by slightly 
more than 50 per cent, in exchange-value in all things other to 
it. And when C has fallen by '-Y-y^ per cent, in A, it has fallen 
by 33^ per cent, in exchange-value in all things, although it has 
fallen by slightly more than 33|^ percent, in all things other to it. 

This is evident, as regards the standard of all things, also if we 
represent the state of things at the second period in these ways, 

B ^ 1 B '-^ 21 C c U A = U D z^ U E ^^ , 

and 

C = IC r^ |B -- |A r |D =^ fE -- ; 

for here it is evident, in the first case, that for B to rise in ex- 
change-value in all things by 50 per cent., it must rise still more 
in exchange-value in at least one of the others in order to counter- 
balance the fact that it does not rise in exchange-value in itself, 
and in the second, that for C to fall in exchange-value in all 
things by 3">i jier cent., it must fall still more in exchange- 
\'alue in at least one of the other things in order to counter- 
balance the fact that it does not fall in exchange- value in iuself.' 
This solution of the difficulty has been anticipated in Proposi- 
tion XXXIII. It is also illustrative of Propositions XII. and 
XIII. For from the variations of B and C in relation to A we 
might gather that B has become 2| times as valuable as C, and 
C ^ as valuable as B. This is true of their exchange-values in 
all things, and in all other things beside them both. But it is 
not true of their exchange-values each in all other things, since 
in all the things other to it B has become somewhat more than 
2^ times as valuable as C has become in all the things other to 
it, and inversely. 

1 It will be noticed that in l)ot.li these eases the extra variation of the one 
other class (equally important over both the periods with the class in question) is 
the square of the common variation of all the other classes. This relation is 
universal. Hereby is given a partial (and in complex cases an approximate) 
answer to a problem proposed in Chapter II. Sect. V. ^ 0. 



CHAPTER XIV. 

THE UTILITY OF MEASURING THE VARIATIONS IN THE 
EXCHANGE- VALUE OF MONEY. 



§ 1, Knowledge of the constancy or variation of the ex- 
change-vahie of money is useful both for theoretical and for 
practical purposes. 

For theoretical purposes it is useful in many scientific en- 
quiries, which lead on to conclusions of great practical impor- 
tance. Thus, for example, there is a prevalent opinion that a 
rise or fall m the exchange-value of money has considerable in- 
fluence on industry and general prosperity, partly deleterious 
and partly beneficial, the one in some ways, the other in others. 
This opinion is to some extent based on experience in flagrant 
instances when there could be no doubt what course the ex- 
change-value of money was taking ; but as yet it has mostly 
been based on reasoning a priori. For it cannot be scientifically 
investigated until variations in the exchange- value of money are 
scientifically determined. Its scientific investigation, however, 
is of the greatest moment ; for if there be truth in the doctrine 
that the deleterious influences are greater than the beneficial, 
and more so of a rise than of a fall, the detailed knowledge of 
such influences may lead to corrective and even to preventive 
measures. Attempt may perhaps be made to attack the source of 
the evil by regulating the exchange-value of money — both to 
prevent the insidious changes of metallic money over long 
stretches of years and the sudden changes of credit money dur- 
ing short periods. For this purpose also knowledge of the 
causes of the variations in the exchange- value of money will be 

472 



TIIEORKTICAL IM'1{I'()SES 473 

nc'ces.^arv. But the prevalent opinions on such causes can like- 
wise he scientifically confirmed or refuted only after more scien- 
tifically accurate measurements of the variations in question 
have heen instituted than any yet made. On the whole, it is 
apjtarent that, as observed by Dana Horton, the theory of the 
multiple standard is "■ the key to the entire theory of money." ^ 
Of the corrective and preventive measures more will be said 
pi'escntly. 

5? 2. Then again we have need of knowing how commodities 
themselves vary in general exchange-value ; for we cannot well 
investigate the causes and consequences of their variations in 
such exchange-value imperfectly measured. To be sure, we can 
easily investigate the causes of the variations of one commodity 
in exchange- value in another, as such variations are })lain. But, 
after all, to know the causes of these particular variations is not 
so important as to know the causes of the general variations, 
which latter have commonly been the subject really had in mind 
when economists have treated of the causes of variations in ex- 
change-value — especially when specifying, as they so often do, 
that they are dealmg with the causes of variations in prices 
under the supposition of money being constant in general ex- 
change-value. Reasoning on this subject needs to be based 
upon experience, and therefore we should be able to convert 
the supposition of money being constant into reality by correct- 
ing its deviations in the instances taken from experience. Other- 
wise a commodity may appear to have risen or fallen in general 
exchange-value because its price has risen or fallen, although its 
general exchange-value may really have varied in the opposite 
direction. Money being habitually used as a measure of general 
exchange- value notwithstanding its own variableness in general 
exchange-value, we need to correct the results obtained from 
measurements with this imperfect instrument, after first meas- 
uring this instrument itself. For after finding its variations 
we can adapt the variations of the prices of commodities to 
variations in general exchange-value, in a manner already ex- 
plained. We may thus obtain what have above been called the 

1 Silver and Gold, 2d ed., p. 40. 



474 UTILITY OF MONETARY MEASUREMENTS 

true-prices of commodities, namely the prices they would have 
had, had money remained stable in exchange- value, and had no 
other changes occurred.^ This term, to repeat, is not inappro- 
priate, because in spite of the variations of money we do con- 
tinue to make use of prices for measuring variations in the ex- 
change-values of commodities, not only in money, but in things 
in general ; but it is only these adapted prices in an invariable 
imaginary money that are true for the latter purpose. 

An example in point may be taken from a subject now agitating 
public opinion. In considering whether the present tendency of 
productive bodies in the same line of business to combine and 
thus avoid competition is beneficial to the country at large, or 
otherwise, one of the items discussed is whether the so-called 
" trusts " already formed have raised or lowered prices. Now 
to discuss whether they have raised or lowered merely actual 
prices is only to discuss whether they have raised or loAvered the 
exchange-value of their products in relation to money. But 
actual experience of mere changes m price of any particular 
class of commodities shows only a change in the relation between 
the general exchange- value of the class in question and the gen- 
eral exchange-value of money ; and does not show whether the 
change is on the side of the commodity or on the side of money, 
or how much on the one and how much on the other — that is, it 
does not show how much the change is due to the efforts of the 
producers of the commodity and how much to the efforts of the 
producers of money. This can be shown — for we are dealing 
with variations, not in cost-value or in esteem-value, but in ex- 
change-value — only by investigating further the relation of both 
these classes to all conmiodities ; which can be done very labor- 
iously in regard to one of the things, preferably money, and 
then very easily in regard to the other. And after doing this 
we are not so much interested in the relative accomplishments 

^ This second proviso is necessary because tlie variation of money in general 
exeliange-value may have had some influence, to be noticed later, to change 
relative exchange-values from what they would be, had money remained stable. 
But the variation having taken place, and exerted its influence, the true-prices 
still indicate the variations of the commodities in general exchange-value under 
such influences. 



THEORETICAL PURPOSES 475 

of the producers of the given eonimodity and of the ])roducers 
of the metal used as money, as in tlie relative accomplishments 
of the producers of the commodity in (piestion and the producei's 
of all other commodities. Hence our interest is really in the 
variations of the commodity's true-price ; for, although such 
variations do not show the commodity's variations in cost-value 
or in esteem-value, they do show the relation between its varia- 
tions in these values and the variations in them of all com- 
modities, — about which relation more will be said in the next 
Section. Yet this aspect of the question is mostly forgotten, and 
the question is often thought to be settled by an appeal merely 
to the actual variations of the prices of the given commodity. 
Thus, for instance, in a recently published work is to l)e found 
the following passage : " The price of cotton-seed oil has fallen, 
along with the economic improvement in its production intro- 
duced by the trust. In 1878 the average price of standard 
summer yellow oil was 47.94 cents per gallon. In I880, the 
year before the organization of the trust, it had only fallen to 
47.08 cents per galh^i. In 1887, four years after the organiza- 
tion of the trust, it had fallen to 38.83 cents per gallon. In 
other words, during these four years the price of cotton-seed oil 
fell more than eight times as much as it did during the five 
years before the trust was formed."^ Here no reference is made 
to the fact that after 1878 industrial conditions took an upward 
swing, which lasted till 1883, and was followed by depression. 
Now if the measurement of the course of the exchange-value 
of money during these years provided by the Aldrich Report 
were reliable, the true-prices of cotton-seed oil, calculated for 
the two later years, the price at the first year being taken as the 
base, would be 44.37 cents per gallon in 1883 and 41.89 cents 
per gallon in 1887. In other words, the fall before the organi- 
zation of the trust was by 7.45 per cent., and the fall after the 
organization of the trust was by 5.52 per cent., so that the true 
fall, instead of being eight times greater in the later period, was 
only three fourths as great.* Unfortunately the index-numbers 

»G. Gunton, Trusts and the public, 18t»9, p. 15. 

* On p. 218 of the same work tlie prices of petroleum are given for a similar 
purpose (the standard Oil Company being established in 1871 and the trust in 



476 UTILITY OF MONETARY MEASUREMENTS 

of the Aldrich Report not only do not refer only to prices in 
New York, where the above prices were reported, but also were 
calculated in an improper manner. Except that the figures of 
the Aldrich Report cannot in most cases be far from the truth, 
we are left in the dark as to the exact movement of the true- 
price of any commodity during the years preceding and succeed- 
ing the organization of its trust ; and to the extent of this ob- 
scurity all argumentation on this subject is confused and con- 
fusing. 

In view of the habitual inattention and neglect with which 
the subject of true-prices is treated not only by the people at 
large but by economists of repute, it is somewhat discouraging 
to find, as recalled by Dana Horton, that the need of observing 
them was pointed out more than two hundred years ago. In 
1672 Pufendorf wrote : "When the price of any one and the same 
thing is said to vary, it must be carefully distinguished whether, 
properly speaking, the value of the thing or the value of money 
has varied.'" And yet perhaps the first, and perhaps even the 
last, writer who has attempted to make a scientific investigation 
concerning true-prices is Professor Laspeyres, who wrote on 

1880). Some of these prices, in cents per gallon, are here given, followed by the 
true-prices calculated from the figures in the Aldrich Report : 

1863 30.7 30.7 

1867 20.5 16.4 

1871 21.7 18.1 

1873 16.0 13.4 

1877 15.0 14.7 

1879 8.125 8.6 

1880 9.125 8.7 
1884 8.25 8.4 
1889 7.125 7.7 
1891 6.9 7.6 

The inferences to be drawn from the latter figures are somewhat diflferent from 
those from the former. In the eight years of open competition the true-price 
fell 41.05 per cent., at the average rate of 4.39 per cent, per annum; in the 
nine years of preponderance of the Company it fell 51.93 per cent., at the average 
rate of 4.75 per cent, per annum; and in the eleven years of quasi-monopoly 
under the trust it fell 12.64 per cent., at the average rate of 1.09 per cent, per 
annum,. 

^ De jure naturce et gentium, lib. Y. cap. I. § 16. Pufendorf, however, 
would have judged the variation in the exchange-value of money by that of 
farm land. Yet the multiple standard was known still earlier, as we shall see 
presently. 



PRACTICAL I'l'RPOSES 477 

this subject about thirty years ago." He deserves credit for so 
doing, although his work was vitiated by a faulty method of 
calculating the variations of money. Other writers have but 
barely noticed the subject,^ 

§ 3. The practical purposes above mentioned are sought to 
be effected in the schemes for correcting the variations of money 
in its function as a measure of exchange-value, and, as far as is 
possible, for preventing such variations. 

To begin with the former. It has been proposed that the 
mensuration of the exchange-value of money should serve as a 
guide in credit operations extending over at least six months, 
or a year, and longer. The design is that, by agreement be- 
tween the parties at the time of contracting, debts of all sorts 
should be repaid in the same amount of exchange-value as was 
borrowed or bargained for, and therefore the sum of money « 
pledged should be paid with addition or deduction according to 
the fall or rise of money in exchange- value. For example, if 
between the time of contracting and the time of solution money 
is found to have depreciated 10 per cent., a person owing 100 
money-units, knowing that 100 of the new units are worth only 
90 of the old units, and that 10 of the old are now worth 

1.00 , , . , 

10 X (.^, = 11.11, must pay back 111.11 money units ; and 

if money has appreciated, instead, by 10 per cent., the same 
debt is discharged by the repayment of only 90.91 money-units, 
the sum due for interest being increased or diminished in the 
same proportion. 

This proposal, as is well known, was revived by Jevons, after 
having been suggested as a serious scheme, perhaps first, by 
Joseph Lowe, near the commencement of the century just 
elapsed, and soon afterwards, in dependence upon Lowe, by 
Poulett Scrope. It has been recommended by several other 

^Especially in his essay Welche Waaren werden im Verlaufe der Zeiteii immer 
theurerf — Statist ischen Studien zur Geschichte der Preisen, in the Zeitschrift fiir 
die gesammte Staatswissenschaft, Tiibingen 1872, Iste Heft. He had previously 
touched upon the subject in B. 25. 

'' E. g., Sauerbeck in B. 79, p. 599. — A. casual use of a solitary true-price is to 
be found in the works of D'Avenel, B. 117, p. 6, B. 118, p. 4. 



478 UTILITY OF MONETARY MEASUREMENTS 

writers, and recently by Professor Laves as something new.* 
Yet hardly new, even when Lowe wrote, was the idea of it, 
which has been before the eyes of every jurist for nearly three 
hundred years. For in the great work of Grotius is the fol- 
lowing passage : " Concerning money we must know that it 
naturally possesses the capacity to pay debts,^ not in its ma- 
terial alone, nor in its denomination, but in a wider respect, 
namely, as it is compared either with all, or Avith the most 
necessary, things ; which estimation, unless otherwise agreed 
upon, is to be made at the time and place of solution." ^'' 

§ 4. The other proposal is that the mensuration of the ex- 
change-value of money should be employed as a guide for 
regulating the currency. A variation of money being detected 
before it has had time to go far, it has been suggested that the 
government can restore to money its former exchange-value by 
increasing or decreasing its quantity, and by performing this 
operation constantly, it can keep money always within very 
slight and inconsiderable deviations from a permanent ex- 
change-value — as the helmsman steers his boat by arresting its 

8 Lowe, B. 8, pp. 278-279, 281-291; Scrope, B. 9, pp. 407-408 (followed by R. 
H. Walsh, B. 13, and reviewed by Maclaren, B. 17); Jevons, Money and the 
mechanism of exchange, pp. 328-333 ; Horton, Silver and gold, pp. 36-43 ; F. A. 
Walkei" (confining it to persons not in business), 3Ioney, pp. 161-163, 3Ioney in 
its relation to trade and industry, pp. 70-77 ; Marshall, in a paper read before 
the Industrial Remuneration Conference, pp. 185-186 of the Report, London 
1885, and in B. 83, pp. 363-365 ; T. Laves, Die " Warenivahrung " als Erganzung 
der Edelmetalwahrung , Schmollers Jahrbuch fiir Gesetzgebung, Verwaltung und 
Volkswirthschaft, Leipzig 1890, pp. 837-846. The scheme is entertained by 
Laughlin, History of bimetallism, 1885, pp. XI.-XII., and Elements of polit- 
ical economy, 1887, pp. 76-77, and by GitFen, Recent changes in prices and incomes 
compared, .Journal of the Statistical Society, London 1888, pp. 54-55 ; and is 
recommended as a substitute, in case of failure to establish the next scheme, by 
H. Winn, The multiple standard, American Magazine of Civics, Dec. 1895, p. 
584, and Parsons, B. 136, p. 333. The present scheme was also favored by Zucker- 
kandl in.B. 115, pp. 249-250 ; but later he has found fault with it for not allowing 
for stability of money in cost-value (or esteem- value) when its exchange- value 
is rising because of improved production of commodities [and not of the money- 
material], B. 116, pp. 249-252. 

^ Functio — fungibleness, the ability of one portion of money to be paid back 
in return for another portion of money borrowed. 

^° De jure belli et pads, 1625, lib. II. cap. XII. g 17. The emphasis in this 
passage lies upon the word "naturally." According to Grotius the multiple 
standard is the natural standard for paying debts, although it has never been 
employed within any State. 



I'lJACTicAL rrui'osEs 471) 

incipient deviations from the true course. The money whose 
quantity is to be regulated has generally been chosen to be 
paper money, issued by the government either directly or 
through the mediation of a bank or banks, and either incon- 
vertible or convertible into a variable amount of metallic 
money or bullion. In the latter case tiiis scheme is somewhat 
like the preceding, except that, the medium of exchange down 
to the smallest subsidiary coins being co-ordinate with the 
largest bills, this scheme will extend to all piiyments even of 
the smallest and shortest contracts, while that scheme was 
recommended only for large and long contracts ; and, again, 
this scheme must from the beginning be compulsory, the paper 
money being legal tender, while in that scheme, the idea was 
that the contraction of obligations payable according to the 
multiple standard should, at least until the practice became 
customary, be voluntary. In the present scheme the sugges- 
tion has sometimes been made that the quantity of the out- 
standing money may be regulated by raising or lowering the 
rate of discount according as prices rise or fall : otherwise the 
regulation of the quantity, by some other method of extending 
and contracting the issues, would be anterior, itself influencing 
the prevalent rate of discount. Such a scheme, more or less 
definitely worked out, has been frequently recommended, more 
or less vigorously. '^ It has even been extended by Professor 
Walras to metallic money in one of its species, namely, to silver 
coins, in a system, as he calls it, of bi//on nr/ii/afrnr, for which 

^^ It was hinted at by R. Walsh, A Letter to A/e.raiid/'r Bariwj, Esq., on the 
present state of -the currency of Great Britain, in the American Review of History 
and Politics, Vol. II., 1811, pp. 275-277, and by Scrope, B. 9, pp. 418-419; and is 
said to have been recommended by W. Cross, Standard pound rersus pound 
sterling, 18oG. A very imperfect form of it was suggested by .Jevons, op. cit., 
pp. 327-328. It has been advocated more seriously by J. Barr Robertson before 
the Gold and Silver Commission, Second Report, 1888, qq. ti294-6304 ; A. Wil- 
liams, A 'fixed value of bullion ' standard. — A proposal for preventing general 
fluctuations of trade. Economic Journal, June 1892, pp. 280-289; J. Conrad, in 
Wissenschaftliche Gutachten uber die Wahrungsfrage, Berlin 1893, pp. 33-34; 
O. J. Frost, The question of a standard of value, Denver 1S94, p. 2fi; Osborne, 
op. cit., p. 332; Fonda, B. 127, pp. 158-195 ; H. Winn, op. cit., pp. 579-589; J. 
A.Smith, B. 129, pp. 33-42; Whitelaw, B. 13(1, pp. 20-22, 2S-S2 ; Parsons, B. 
136, pp. 102 ff.; T. E. Will, Stable money, Journal of Political Piconomy, Chicago 
Dec. 1S9S, pp. 85-92. 



480 UTILITY OF MONETARY MEASUREilEXTS 

the existence of " limping bimetallism " in many countries offers 
opportunity.^^ 

§ 5. A warning should be given to the advocates of either of 
these schemes. This is that there is no such thing as a single 
variation in the exchange-value of money, or of gold or silver, 
the same throughout the whole world, or between adjoining 
countries, especially if separated by shifting tariff barriers, or 
even within the borders of any one fairly large country. For 
the changes in the charges of transportation and of intercourse 
cause the variation in the exchange-value of the same mone- 
tary system to be different in different regions. In our coun- 
try, for instance, the variations in the exchange-value of money 
at IS^ew York, at New Orleans, at Chicago, and at San Fran- 
cisco, would form four appreciably different series. Each of 
these series should be measured by itself, no prices in one part 
of the country being mixed up with prices in another part. 
Then, in the former voluntary scheme, the contractants could 
use the index-numbers of their own locality. But in the second 
compulsory scheme, as no one center alone should be favored, 
the standard for the whole country should be an average of 
the variations at the different centers, each being weighted 
according to, not so much the population, as the total wealth, 
of its region. 

On the other hand, to the opponents of any such schemes 
may be given an admonition. There is a not uncommon opin- 
ion, to be found even in the works of very respectable econo- 
mists, that the exchange-value of metallic money, and among 
English-speaking peoples, of gold money, is normally stable at 
certain levels, and that periods of variation are only transitions 
from one level to another. Accordingly, when nations have 
felt themselves suffering from monetary appreciation or de- 
preciation, assertions have confidently been made that only a 
little patience is needed, as the condition is transitory and will 

12 In the Journal des Economistes, Dec. 1876, May 1881, Oct. 1882; B. 71, p. 
11; B. 69, pp. 2-3, 12, 16-18 ; B. 70, pp. 411,441-449, 478-484; B. 71, pp. 143-144, 
148-151, 162-163, 484.— Walras is followed by Andrews, B. 107, pp. 36-46, B. 108, 
p. 141 ; and was partially anticipated by Mannequin, La monnaie et le double 
etalon, 1874, p. 59. 



I>I!IX('II>J>ES OF CUKKENrY l{E(;ri>ATl< )N 4!S] 

soon be followed by a settled condition when money shall have 
found its new lev^el ; and therefore to do anything now will 
only be to disturb that future happy state soon to be entered 
upon naturally. Nothing could be more false. There are no 
periods of stability in the exchange-value of metallic money. 
No measurement, among the measurements as yet made, of the 
course of the exchange-value of money, has ever shown any- 
thing but perpetual movement. To be sure, it sometimes has 
happened that the upward and downward courses are short and 
nearly counterbalance one another over an epoch of several 
years. But such occasions have been the exception, and not 
the rule. The so-called industrial cycles of eight or ten years 
have rarely been at the same average level of prices. The 
rule is rather that the fluctuations are uneven, and have a rising 
or falling tendency for many years at a time, the durations of 
these epochs themselves being various, so that anticipation is 
impossible for more than a year or two in advance, and then 
with great uncertainty. Natural money has no permanent level 
of ex change- value ; nor can we even rationally hope for such 
a level. Perhaps, when the variations in exchange-value both 
of money and of commodities, the latter by means of their 
true-prices, have been better investigated, it will be found that 
the precious metals are among the products of human industry 
the most variable in exchange-value. 

II. 

§ 1. The principles upon which rests the scheme of regu- 
lating the exQhange-value of money by exercising artificial con- 
trol over its issuance, deserve our further attention. In this 
work, however, we are not concerned with the investigation of 
causes ; and so the quantity theory of the cause of the exchange- 
value of money, which is the foundation for that scheme, is not 
to be discussed here. It may be assumed for the sake of argu- 
ment. To be examined are mere relations between prices and 
exchange-values. 

In the preceding pages the question has been investigated as 
to what price variations ought to take place in order to com- 
31 



482 UTILITY OF MONETARY MEASUREMEXTS 

pensate for given price variations, the mass-quantities or the 
money-vahies of the classes also being given. Now in the 
actual world, when one article rises a certain percentage in 
price, it is impossible by any manipi*lation of the quantity of 
money to effect a fall of another article alone, or of all other 
articles, to exactly the proper extent for counterbalancing the 
given rise. For any manipulation of the quantity of money, if 
it aifects prices at all, affects not only the price of the other 
article, but the prices of all other articles, and not only these, 
but the price of the first article itself, and so destroys the 
datum. We have, however, the following principles. 

Commodities change in price relatively to one another according 
as they change in exchange-value relatively to one another, which 
latter relationship is hardly affected by changes in the quantity 
of money. ^ But changes in relative exchange-values may mani- 
fest themselves in prices in an infinitude of ways. For instance, 
if we had the articles [A] and [B] in certain quantities, A and 
B, each priced at one money-unit, and therefore equivalent, and 
later they change relatively to each other so that A ==- IJ B, 
this could be (1) by A rising in price to 1.50 and B remaining 
at 1.00, (2) by A remaining at 1.00 and B falling to .66f , (3) 
by A rising to 1.20 and B falling to .80, (4) by A rising to 2.00 
and B rising to 1.33^, (5) by A falling to .75 and B falling to 
.50, or in any of numberless variations upon the last three 
combinations. Which of these shall he the one actually to take 
place, depends upon the relationship later existing between [J.] 
and [5] together [along loith all other commodities) on the one 
hand and money on the other.^ 

1 Hardly, because there is some influence : (1) there is a tendency in prices to 
round numbers, so that they change, not continuously, but discretely — "by 
hitches and starts," to borrow a phrase from F. Bowen, American political econ- 
omy, p. 411 ; (2) noticeable changes in the quantity of money, differently affecting 
the prosperity of different producers, influence some articles sooner than others in 
what A. Del Mar has felicitously called a " precession of prices," The science of 
money, London 1885, Chapt. VIII. 

2 Money is thus contrasted with all other things, not because of its unique 
nature already referred to, but merely because its exchange-value in all other 
things is under consideration. If we were considering the variation of any one 
commodity class in general exchange-value, we should similarly have to contrast 
it with all other things. 



PRINCIPLES OF criMiKNCY 1!E< iTLATK >X 483 

Here, again, in our conception of the actual variations in the 
exchange-values of the commodities, we are no more concerned 
with the causes of these variations than we have been hitherto. 
[B] may fall in exchange-value in [A] because [H] is now 
produced more cheaply and abundantly than formerly, while 
[A] is still produced with the same difficulty and in no greater 
quantity, or because [B] is still produced with the same diffi- 
culty and in the same quantity, while [A] is produced with 
greater difficulty and in lesser quantity, or because of three 
more typical combinations of such changes ; or again because 
the demand for [B] lias diminished, while that for [A] has 
remained the same, or because the demand for [B] has re- 
mained the same while that for [A] has augmented, or because 
of three more typical combinations of such changes — making 
five in each kind, which may act singly or in unison, thus pro- 
viding us with as many as thirty five typical combinations of 
causative changes. But these causes, so far as operating be- 
tween [A] and [B], or between these and all other commodi- 
ties, even if perfectly and completely known to us, in no wise 
inform us what are to be the changes in the prices of [A] and 
of [B] , or explain to us why they are such as they are, except 
as regards their relation to each other ; for these changes are in 
relation also to money, and depend upon a similar interaction 
of causes (again in thirty five typical combinations) operating 
between [A] and [B] and all other commodities on the one 
hand and money on the other. In other words, without tiie 
factors affecting money, the factors affecting commodities can- 
not alone determine prices ; nor the former alone without the 
latter. The two sets of factors cannot operate independently 
of each other. Of course the changes in the causes may be in 
only one of the two sets, the other remaining untouched. But 
in whichever set it be, the effect upon the exchange-value or 
exchange-values on the one side is, ipno facto, an inverse effect 
upon the exchange-values or exchange-value on the other. 
For the exchange-value of money is affected as well by the 
causes affecting the production and quantity of commodities as, 
inversely, by the causes aff^ecting the production and quantity of 



484 UTILITY OF MONETARY MEASUREMENT 

money ; aud the level of prices is only a manifestation of the 
effect, in whichever way it be brought about.^ Therefore the 
residence of the changes in the causes on the one side or on the 
other should have no influence upon our opinion concerning 
the variation or constancy of the exchange-value of money. 
This is to be decided only by a measurement of the actual con- 
ditions, — most conveniently by the measurement of the inverse 
variation of the general level of prices. Changes in the causes, 
moreover, if taking place on both sides simultaneously, and 
influencing both sides in the same direction, may neutralize 
each other and so leave the exchange-value of money constant. 
The exchange-value of money will be altered only (1 and 2) 
by the existence of such changes on either side alone, either 
with an upward or with a downward influence, this influence 
being neither aided nor impeded by an opposite or by a similar 
influence coming from changes on the other side ; or (3) by the 
influences of such changes on both sides in opposite directions, 
either upwards for commodities and downwards for money or 
downwards for commodities and upwards for money, these 
influences combining to increase the divergence ; or (4 and 5) 
by an excess of the influence of such changes on either side 
alone operating in either direction over the influence of such 
changes operating in the same direction on the other side. And 
of course what is here said of the general exchange-value of 
money may be said of the general exchange-value of any class 
of things over against all other things. 

§ 2. Now what everbody wants in regard to all commodities 
is that they should individually and collectively become 
cheaper in cost-value and in esteem-value. Many persons, 
then, have inclined to think that what is desirable is that all 
commodities should become individually and collectively cheaper 
in exchange-value also. Here is a mistaken inference due to 
confusion of thought, itself due to the confusing use of a single 
term " value " (and of the allied adjectivial terms " dear " and 
" cheap ") to express what really are distinct ideas, the genus 
being used for the species because the species have not been 

■•» See above. Chapt. II. Sect. III. § 1- 



PRINCIPLES OF CURRKNCV REGULATION 485 

sufficiently distinguished, in the error which logicians call the 
fallacy of undistributed middle.' 

The inference is mistaken, as is plain from the fact that it 
violates Proposition LII., and what is inferred is impossible. 
To renew the proof of this, let us suppose that the alleged de- 
sire for the cheapening of everything in exchange-value is sat- 
isfied by the successive equal cheapening of everything. If 
[A] becomes cheaper in exchange-value first, ip.^o facto, as we 
know, everything else becomes slightly dearer in exchange- 
value. If later [B] becomes equally cheaper in exchange- 
value, again everything else (including [A]) becomes slightly 
dearer in exchange-value. Also [B] is then as cheap as [A] , 
wherefore [A] is as dear as [B]. Again, if [C] becomes 
equally cheaper in exchange-value, everything else (including 
[A] and [B] ) becomes slightly dearer in exchange-value ; and 
now [A], [B] and [C] are equally cheap and equally dear. 
And the successive fall of everything else will tend to raise 
slightly the exchange- value of everything else (including the 
things already fallen). Every single thing, therefore, at 
some time falling in one jump the full extent of the common 
fall supposed, and then effecting a slight rise in all the others, 
will itself be raised in very small stages by the falls of all the 
others, whether before or after, or partly before and partly after, 
its own fall. Thus when all but one have fallen, that last thing 
will be dearer in exchange-value than it was at the commence- 
ment by the inverse of the fall of all the rest. When it too 
falls, the last rise in the others will be consummated, and as the 
last thing has fallen back to equivalence with all the rest, [A] 
and [B] and [C] and [D] and the rest through the whole list 
down to the last one now being equivalent, their exchange- 
values will be in exactly the same condition as they were at the 
commencement, that is, everything will have the same exchange- 
value as before, and the very supposition by which everything 
is made successively to fall equally in exchange-value shows 
that when the operation is completed nothing is fallen in ex- 

*Notice that we do not want things to fall in use-value. Such a fall would 
mean an approach, on our part, to insensibility. Rather we would prefer that 
things should rise in use-value. 



486 UTILITY OF MONETARY MEASUREMENTS 

chauge-value. Of course if we should attempt to suppose that 
all things fall equally in exchange-value together and at once, 
the same result would be involved, which means that nothing 
is fallen, although all are supposed to have fallen, so that such 
a supposition is a contradiction in terms. And if the things are 
all supposed to fall in exchange-value unequally, or if only 
some of them are supposed to fall, the supposition remains true 
that these latter are fallen, or, in the former case, that those 
which were supposed to fall most, still are fallen. But at the 
same time it is true that the things supposed to fall least are 
risen in exchange-value. It is impossible to carry out the con- 
ception of all things falling together, equally or unequally, in 
exchange- value. 

The reason why any difficulty should appear in this subject 
is that it is possible for people to make the supposition, or to 
entertain the thought, that some or all things have fallen in 
exchange-value, and at the same time to forget, or perhaps 
never to perceive, that every fall they suppose in exchange- 
value involves also a rise in exchange-value. Of course, then, 
while entertaining the one thought, and excluding the other, 
they may thinlc of all things falling in exchange-value. But 
they can do so only because they are thinking imperfectly. 

And, to repeat, the reason why people have thought so im- 
perfectly is that they confound exchange-value with cost-value 
or with esteem-value ; for in these latter kinds of exchange- 
value it is possible for everything to fall, and to remain fallen, 
the fall of one thing in such values not having any necessary 
influence to raise any other things in such values. A passage 
which may serve as a locus dassieus of this confusion of 
thought is the following, taken from the PoUtical Economy of 
Malthus. In it, to be sure, Malthus uses the term " exchange- 
able value " ; but it must be remembered that he had no terms 
whereby to distinguish from exchange- value the other kinds of 
value except use-value, so that by " exchangeable value," as by 
"value" itself, he variously referred to all the kinds of value 
except the last. Now after supposing a case of equal improve- 
ment in the production of all goods, he proceeds to ask and to 



I'KINCII'LKS OF crifHENC'V I{K(;ULATION 4H7 

answer as follows : — " Can it be asserted witli any semblance of 
correctness, that an object which under these changes would 
command the same quantity of agricultural and manufactured 
products of the same kind, and each in the same proportion as 
before, would be practically considered by society as of the 
same exchangeable value ? On the supposition here made, no 
person would hesitate for a moment to say, that cottons had 
fallen in value, that linen had fallen in value, that silks had 
fallen in value, that cloth had fallen in value, etc., and it would 
be a direct contradiction in terms, to add that an object which 
would purchase only the same quantity of all these articles, which 
had confessedly fallen in value, had not itself fallen in value."'' 
Here, of course, if by " value " be meant cost-value, or esteem- 
value, the conclusion would be perfectly correct. And here 
Malthus himself did have in mind a mixture of cost-value and 
esteem-value," and his confusion consisted in still calling it 
" exchangeable value," which nine-tenths of his readers, and 
himself too at times," take to mean what the term itself prop- 
erly means. His conclusion, then, is both correct and incor- 
rect. For the reasoning is perfect on condition that the falls 
of all the separate classes of things are independent of one 
another, and that there are no rises really involved, but sup- 
pressed. Such is the case with the falls in cost-value, as also 
in esteem-value (or at least to some extent, — that is, the condi- 
tion is a possible one). But such is not the case with falls in 
exchange-value. With regard to this, the appearance of cor- 
rectness in the reasoning is acquired only by suppressing the 
counterbalancing rises necessarily involved in every fall, wholly 
forgetting or ignoring the interdependence and correlation of 
all variations in this kind of value. 

§ 3. The desire, then, that all things should become cheaper 
in exchange-value is impossible, absurd, and inane. But there 
is another confusion of thought possible in this connection, and, 
because possible, often fallen into, which is not so empty and 
harmless. Variations in prices are variations in exchange- 

•'' Second ed., p. 58. 

« Cf. ibid., p. 60. 

' Cf. ib><L, pp. 50 and 61. 



488. UTILITY OF MONETARY MEASUREMENTS 

values ; and therefore variations in exchange-values have been 
confounded with variations in prices. And as people have en- 
tertained the contentless desire that all commodities should fall 
in exchange- value, they have identified this with the desire 
that all commodities should fall in price. Or perhaps this 
desire has been reached directly from the sound and respectable 
desire that all commodities should fall in cost- value and in 
esteem-value, since variations in the latter are frequently, and 
in isolated cases almost always, accompanied by variations 
in the former. Here is something possible. But it involves 
something else, which also is often ignored. This is that if 
every commodity becomes cheaper in price, that is, in exchange- 
value in money, money becomes dearer in exchange-value ; and 
if the former is still desired when this is perceived, the latter 
must also be. Why are these two things to be desired ? The 
only possible reply as regards the first is that the desire for 
the fall of prices is conjoined with the desire for the fall of all 
commodities in cost-value or in esteem-value. It is thought, 
with or without good reason, that the desired fall in these 
values, if occurring, should be marked and measured by a cor- 
responding fall in their prices. And this thought necessarily 
involves the idea that money is to be considered the standard 
measure, not of exchange- value, but either of cost- value or of 
esteem-value. 

It is not within the p'^ovince of this work to argue as to 
whether money should be considered the standard of the one 
or of the other of the various kinds of value. But it is essen- 
tial here to point out that money cannot possibly be the standard 
of more than one kind of value. With but the rarest excep- 
tions, all the world has hitherto been agreed that money is a 
measure of " value," and consequently that in order to serve 
this function properly and to be worthy of being taken as a 
standard, it ought to be stable in " value." It becomes neces- 
sary now to distinguish in which sense the term " value " must 
be understood in this connection. To continue speaking of 
money merely as the measure of " value " will be to perpetuate 
an equivocation of thought that may lend aid to either of 



PRINCriT.ES OF cniRENCY REGT^LATIOX 489 

opposite sides on many important practical questions according 
to the desires selfishly uppermost in the minds of partisans. 
Here is a point which all economists should pause to decide 
and settle, before they attempt to take a further step in their 
science — or at least in that branch of it which is the science of 
money. 

>^ 4. As, however, this work specially deals with exchange- 
value, we may here assume, for the sake of theory, that money 
is properly the standard measure of exchange-value. Then 
money ought to be stable in exchange-value ; and if it is not 
so naturally, it ought to be made so artificially, if this be pos- 
sible. And, as another statement of the same thing, the gen- 
eral level of prices ought to be constant ; and if it is not so 
naturally, it ought to be made so artificially, if this be possible. 
And this in spite of, and through, the fact that it is desirable 
for all commodities to fall in cost-value and in esteem-value. 

The collective will of the people, as organized in government, 
cannot properly control the causes at work upon commodities, 
except for the purpose of unfettering and aiding production, 
and of making all commodities fall in cost- value aud in esteem- 
value.'' In fact, the fall in cost-value is the aim of every 
producer of every class of commodities, while the fall in esteem- 
value, aimed at by the consumers, is the result of the free play 
of all these agencies.'' But it is believed to be within the power 
of government, by assuming the issuance of money, to control 
the exchange-value of money in all things, — that is, also, the 
exchange-value of all things together in money, — without in 
any wise seeking to control their relative exchange-values 
amongst themselves. If this be so, the aim should be, neither 
to make money cheaper in commodities, that is, to make prices 
rise, nor to make commodities cheaper in money, that is, to 
make prices fall, but to keep money stable in exchange- value in 

" Properly,— agsiin explanation is needed, — because as a matter of fact govern- 
ments do exactly the opposite and raise the cost-value and esteem-value of goods 
— by taritfs, keeping out the goods produced more cheaply abroad that could be 
exchanged for the goods produced more cheaply at home. 

'•> Without this free play the producers may succeed, through monopoly, not 
only in lowering the cost-value, but even in raising the esteem-value of their 
goods, by curtailing production. 



490 UTILITY OF MONETARY MEASUREMENTS 

commodities and commodities as a whole stable in money, that 
is, to make the general level of prices constant, so that, while 
the esteem-values of all commodities are happily falling with 
the fall in their cost-values, the esteem-value of money shall 
fall neither more rapidly nor more sluggishly than the esteem- 
values of all commodities on the average. 

Take, for instance, the case above supposed of one commod- 
ity rising 50 per cent, in exchange value in another, the latter 
then falling 33J per cent, in exchange-value in the former. 
Let us revive our earlier supposition of a simple economic 
world with money and two classes of commodities. Now if the 
rise of [A] by 50 per cent, in [B] were manifested in prices 
by [A] rising 50 per cent, in price while [B] remained con- 
stant in price, this would mean that the average of prices has 
risen and the exchange-value of money has fallen ; which is a 
sign that money is too abundant, whatever were the other causes 
at work. Some money, therefore, should be abstracted from 
the circulation. This would have the effect of lowering the 
prices of both the commodities in (about) the same ratio. If 
the contraction should proceed until the price of [A] stood at 
1.2247 above what it was at the start and the price of [B] at 
.8165 below what it was at the start, we know by our investi- 
gations that then, on the supposition of the two classes being 
equally important over both the periods together, the exchange- 
value of money is the same as it was in the beginning. On the 
other hand if the relative change between [A] and [B] had 
manifested itself by [B] falling in price by 33 1^ per cent, while 
[A] remained constant in price, this would constitute a fall of 
the average of prices, indicating a rise in the exchange-value of 
money, signifying an insufficiency of money, whatever be the 
other causes of these changes, and therefore giving warning of 
the need of issuing more money for the purpose of running the 
price of [A] up to 1.2247 and the price of [B] up to .8165 
compared with what they were at the start. Or the prices may 
originally have changed in any of three other typical ways. 
But whatever their original changes, due to natural causes, as 
soon as it is discovered that they are such as to constitute a fall 



PRIXriPLES OF CUUPENfY I!K(;rr.ATI()X 491 

or a rise in the exchange- value of money, the issuance should 
be altered so as to make both the prices move into their proper 
counterbalancing })ositions. 

If one of these classes were more important tiian the other, 
their counterbalancing positions would be diiferent ; but these, 
too, can be calculated, and, when calculated, their attainment 
should be aimed at in the regulation of the currency. Unfor- 
tunately, however, in this case, unless the mass-quantities are 
constant, the calculation cannot be made with absolute preci- 
sion. Also this is so when we are dealing, as we must in our 
complex work], with many and variously large classes. Yet 
we know that still, the more complex the world, the more is the 
use of the properly weighted geometric average of the price 
variations likely to approximate to the truth. Therefore it is 
still possible to used the geometric method with practically 
sufficient accuracy. 

Thus in general, in any complex economic world, if one 
class of commodities rises p per cent, (in hundredths) in ex- 
change-value in all other commodities, then, for money to 
remain stable in exchange- value, money should rise in ex- 
change-value in the other commodities from unity approxi- 
mately to the geometric average between 1 repeated r times and 
1 -|- p, which is ' ^^1 -f p, or, to use another form of expression, 

(1 +^) ' + 1, (/• representing the number of times all the other 
commodities are more important over both the periods com- 
pared than the one class in question) ; wherefore the prices 

of these articles should fall approximately bv 1 — ., '_ 

^^ -^ ' (l+jj).-i 

per cent., and, as the price of the one article must rise to 
\ -\- p times the other prices, it should rise approximately to 

. ^- \_ , or (1 + p)) '■ + '^, times its former price, approximately 

by (1 4- p) 'in — 1 per cent. And reversely, if one class falls 
p' per cent, in exchange-value in all other commodities, its 
price should fall, not by p' per cent., but approximately by 

1 — (1 — p') +1 per cent., and the prices of the others, in- 



492 UTILITY OF ^[OXETAEY MEASUREMENTS 

stead of remaining unchanged, should rise approximately by 

jz. TV J_ — 1 per cent. Or, further, if out of all classes of 

(1 — p')'-i-i ^ ' 

commodities n" in number, certain classes I in number (in both 

cases including the repetitions needed to represent their relative 

sizes over both the periods) rise in exchange-value in all other 

commodities evenly by j) per cent, each, or unevenly so that 

the properly weighted geometric average of their rises is _p per 

cent., then money should rise in exchange-value in the others 

approximately to the geometric average between 1 repeated 

I 
n" — I times and 1 + p repeated I times, that is, to (1 + p) ^' 

times its former exchange-value in them ; wherefore they 
should fall in price evenly or on the average approximately by 

1 — Tz. r i per cent., and the / articles should rise in price 

(1 +p)n"^ ^ 

I 

evenly or on the average approximately by (1 -|- p>y~n'^' — 1 per 
cent. And reversely if the I articles fall p' per cent, in ex- 
change-value in the others, money should fall in the latter 

I 
approximately to (1 — ^j')^' of its former exchange-value in 

them ; wherefore their prices should rise approximately by 

1 

/, TT 1 — 1 per cent., and the prices of the I articles should 

(1 — p')n" ^ ' ^ 

I 
fall approximately by 1 —(1 — p^f'^i^' per cent. Therefore in 

any of these cases, whether addition or subtraction of money 
be needed, it should stop when these prices are obtained.^*^ 

The actual operation of the system, however, would be some- 
what different, and simpler, though less precise. We never 
know directly how much one or more articles vary in exchange- 
value in another or others ; but we learn this only from their 

I'^In practice things would uot work so smoothly as in the theory. Prices 
would not be raised or lowered all in exactly the same proportion, for the reasons 
stated in Note 1. For instance, small alterations in the quantity of money might 
perhaps raise or lower some prices which were just on the verge of making the 
jump from one round figure to the next, while others at the verge of falling 
might merely be advanced to the verge of rising, or reversely, without showing 
any actual variation. Still the principle is the same, and the additiort or sub- 
traction of money should stop when the prices, whatever they be, approximately 
compensate. 



I'RINCIIM.HS OF < TltRENCY I{i:(;i'LATIOX 493 

prices already formed under causes coming from money as well 
as from the commodities themselves. And we do not generally 
know the causes which make the changes either between the 
commodities or between these and money ; nor for our present 
purpose is it necessary that we should know them. Their 
eflPects are manifested in the subsequent prices, and to know 
these is sufficient. Here is where the formuhe for finding the 
new level of all prices compared with the former level come 
into practical use, — and now one of the other two is preferable 
to the geometric. But perfect precision in regard to the exact 
extent of a variation upward or downward is not of great im- 
portance, provided the result be always given in the right di- 
rection. If the level of prices is shown to have risen, it is a 
sign that the quantity of money is too great and should be 
diminished ; and if the level of prices is shown to have fallen, 
it is a sign that the quantity of money is too small and should 
be increased. Of course it is well that the measurement should 
show whether the rise or fall is great or small ; for then it will 
be known whether a great or a small alteration needs to be made 
in the quantity of money." The alteration must continue until 
a subsequent application of the formula shows that the original 
level is again obtained. The possibility of any great variation 
in the level of prices should be cut off by making the measure- 
ments at short intervals. ^^ 

^1 Walras once thought that, the variation of tlie level of prices and the quan- 
tity of money in a country lieing known, he could calculate exactly tlie new 
quantity of money needed, his conclusion being that the former quantity should 
be multiplied by the reciprocal of the variation, B. liO, p. 17. But things do not 
work with such exactness because (I) the principle upon which his calculation is 
founded, namely that, given the quantity of goods, the level of prices is exactly 
in inverse proportion to the quantity of money, is not true, and (2) even if it 
were, there would be intervening changes going on in the goods before the addi- 
tion or subtraction of money could be carried out or before its influence could be 
felt. Walras has since modified his views on this subject, in B. 70, p. 476. 

^- Walras wants the determining measurements to be made for every de<!ade, 
i. €., for an industrial cycle, B. (31, pp. i>, 18 — this is too long. At the other ex- 
treme Williams, op. cit., p. 289, Fonda, B. 127, p. !(», and Whitelaw, B. 130, pp. 
23, 51, want the measurement to be made daily — this is too short. They were 
recommended yearly by Lowe, B. 8, p. 279, half-yearly or cjuarterly by Laves, 
op. cit., p. 842, and Pomeroy, B. 135, p. 333, and monthly by Jevons, op. cit., p. 
330, and Parsons, B. 13(3, p. 137. Perhaps it would be best to make provisional 
measurements monthly or quarterly, and definite measurements yearlj'. 



494 UTILITY OF MONETARY MEASUREMENTS 

§ 5. On the other hand, let lis for a moment suppose that as a 
measure of " value " money is to be considered the measure of 
cost-value or of esteem-value. Then either the payment of 
contracts may be regulated by the ascertained variations of 
money in the one or the other of these kinds of value, or, 
the preventive system being adopted, the aim to be held in 
view would be, by a similar regulation of the quantity of the 
currency, to keep money stable in one of these values, forcing 
the prices of all commodities individually to rise or fall accord- 
ing as the commodities individually rise or fall in cost-value 
(although here there would be difficulty), or in esteem-value. 
Cost-value being selected as the norm, the cost of production of 
the money-metal (for here the cost-value of paper money is out 
of the question) ought on the average to be constant.^^ In the 
case of esteem-value being chosen, the gross- money earnings of 
an hour's labor of all the workers in a country (or their net 
money-earnings, along with the money-incomes of an hour, in a 
day of the number of hours the others work on the average, of 
all the non-workers) ought to be constant. The former of these 
standards has never been suggested either as guide for the pay- 
ment of money -contracts or as guide for regulating the issuance 
of money — the last being possible only by interference with the 
working of the mines of the precious metal or metals, which has 
never been recommended in this connection. The latter stand- 
ard has been suggested for both these uses. The suggestion 
has been made both that contracts should be payable in the 
quantity of money that is constant in esteem-value (imperfectly 
measured by wages only),^^ and that the issuance of money 
should be regulated so as to keep money stable in such value 
(likewise imperfectly measured by wages only).^^ In either 
case, as remarked in an earlier chapter, after having made such 
measurement in regard to money alone, if we have a desire to 
know whether, and how much, commodities in general have 
risen or fallen in cost-value or in esteem-value, we could calcu- 
li On the average, and not at the least fertile mine, because we are now not 
dealing with relations of exchange or of esteem. 
1* By Shadwell, Principles, p. 260. 
^5 Both indifFerentlj% by Pollard, op. tit., pp. 74-75. 



principlp:s of (Tuukntv i!K(;iLA'ri(>N 4JI") 

late this (in cost-value only imperfectly) by inverting the cal- 
culation of the variation of such money in exchange-value in 
all commodities. Thus in these cases also the measurement of 
the general exchange-value of money would still be useful. 
Only now its use would be merely theoretical and to satisfy an 
idle curiosity, while the really important measurement would 
be the measurement either of cost-value or of esteem-value. 
But, on the contrary, if money is properly the measure of ex- 
change-value, it is these other measurements that are practically 
useless and serve only to please our vanity by showing that we 
are making progress. 

§ (). Both the above discussed jn-actical schemes for making 
use of the measurement of variations in the general exchange- 
value of money, or in the general lev^el of prices, are still em- 
bryonic, and no attempt to apply them will probably be made 
for centuries to come."^ But a stumbling block in their way is 
the fact that the measurement of exchange- value has never been 
perfected so as to win unanimous assent on the part of econ- 
omists, as is necessary before scientific knowledge can be claimed 
concerning the very variations which those schemes propose to 
correct or to prevent. Also this lack of science, which is still 
greater in regard to cost-value aud to esteem-value, is no doubt 
a reason why economists have not turned their attention to a 
more careful consideration of the question whether money is the 
measure of exchange- value or of cost-value or of esteem-value. 

I'^Of course the mensuration of exehange-value should not be postponed until 
practical use can be made of it ; for practical use can be made of it only after it 
has been performed. Hence it is advisable even now that in the mass of statistics 
which every civilized government makes it a duty to collect there should be an 
effort made to measure the variations in the exchange-value of money. The cre- 
ation of an official bureau for this purpose has been recommended by Scrope, B. 
9, p. 424; Newcomb, B. 76, p. 213; Marshall, B. 93, p. 365; British Association 
Committee, Fourth Report, B. 102, p. 487 ; Laves, op. cit., p. 846 ; Zuckerkandl, 
B. 11.5, p. 249. 



APPENDIX A. 

ON VARIATIONS OF AVERAGES AND AVERAGES OF 
VARIATIONS. 

The supposition is that we have two sets of figures, ttj, b^, , at a first 

moment or period, and at a hiter moment or i>eriod certain other figures, 

Oj, b^, , to which those have variedJ We may liave any numbers of 

these figures at both periods, hut always tlie same number in each set ; for 
otlierwise we should have a figure in the first wliich does not vary to anything 
in the second, or a figure in the second to which nothing in the first has varied. 
The number of the a's, whether one or more, may be I'epi'esented by /, tliat 
of the 6's by //, and so on. For convenience, although this phraseology is not 

strictly accurate, we may speak of the symbols a^, (tj, b^, b.^, , as figures, 

and of the symbols/, y, , as numbers (this referring to the numbers of 

times the figures occur in each set). The figures constitute, so to speak, 
classes, there being as many classes as there are distinct figures. We may 
have reason, provided by the problems we are dealing with, to divide even 
similar quantities into distinct classes, and tlien to represent them by distinct 
figures.* 

The figures, of course, are quantities. But sometimes it will be convenient 
to speak of their si'scs, when it is desired to call attention to their quantitative 
relations. The classes also have sizes, made up of their figures multiplied by 
their numbers, — e. _</., xa^, .ra.^, ybi, etc. 

The variations of the figures from tlie first to the second period are repre- 
sented by the expressions ^, '^ , The variation of n, to «., is a variation 

«i ^1 

of a, by ■^ ; for a, • ' - u.,. It is the same also as a variation of 1 (= ' 1 to 
' -^ «,■'«, - V <hJ 

' . Of coui-se the variation of the whole class in, to xa.^ is nothing else as a 
a^ 

variation ; for ^^r= '^ . But the variations of all the figures in a class, thus 
.rai Oi 

^ If «2 =«i. this may he viewed as an infinitely small variation. Therefore 
we may include it in speaking of variations. 

- If «! = bi, but a., and b-, are unequal, or if a, = 62. but «! and />i are unequal, 
this is already sufficient mathematical reason for putting the quantities, though 
the same at one period, into different classes. But even if «i = /', and aj = b.^, 
there may he other reasons for treating them as belonging to distinct classes. 
Though classes properly contain many individuals, single figures without mates 
may still be viewed as constituting classes. 

32 497 



498 APPENDIX A 

-, -, to 2: terms, when added together, are x — , or when multiplied by 

one another, are f — ) . 

We may average each set of figures separately. Representing the average 
of the figures at the first period by Ai, and that of the figures at the second 

A 

period by A2, we may represent the variation of these averages by ~ , the 

variation departing from constancy as this expression departs from unity. Or 
Ave may average the variations of the figures as they change from what they 
were at the first period to what they become at the second. Eepresenting this 

A 

by V, we shall find sometimes tliat x^ = V, and sometimes that these expres- 

sions are different. It is our purpose to study the relationship between these 
forms. 

In averaging, to assign a certain importance to one term — a class of figures, 
or a kind of variations — relatively to others, is to weic/ht it. The most natural 
way to weight classes of figures is by the numbers of times the figures occur 
in each class. Similarly it would seem most natural to weight kinds of vari- 
ations by the numbers of variations in each kind. But as the figures may be 
reported capriciously, and the steady element be the sizes of the classes, we 
shall find the need of weighting the variations according to the latter, in 
several ways. 

The number of all the classes, and consequently of all the kinds of vari- 
ations, will be represented by n, so thatn^ 1 for Qj, a^, or '" -j- 1 for 6,, 
b.,, or v^ 4" through the whole lists. The number of individual fig- 
ures at each period, and the number of the individual variations, will be rep- 
resented by n^, so that n^ =^ x-\- y ■-{- to n terms. If there is only a single 

figure in every class, then ?i^ := n ; otherwise n^ > 11. Thus n^ represents the 
sum of what we have just described as the most natural weights of the classes 
of figures we are dealing with. The sum of the other weights which we 
shall need for the variations (and sometimes for the figures) will always be 
represented by n^% although these weights are different on different occasions. 

Before proceeding to our subject proper, it may be well to state certain 
p rinciples which are common to all kinds of averages. 

I. ON AVEEAGES IN GENERAL. 

I 1. An average of any set of terms is alivays smaller than the largest and larger 
than the smallest, whatever be the number of terms. Consequently 

^2. If the terms are all equal, the average is equal to each term, whatever be 
their number. 

These two principles are explicative, that is, they flow from the defini- 
tion of "average," and no formula pretending to be a formula of a kind of 
average is so unless it yields results in accordance with these principles. That 
the formulfe for the arithmetic, harmonic and geometric averages satisfy the 



VAIMATIOXS AM) AVHJJAOKS 499 

second principle lias incidentally l)een shown in Chapter V. Sect. I. §<i 5 
and 6. 

Consequently also, if we are dealinji; with sets of terms in classes, 

§ 3. The average is always between the highest and tlie lowest terms appearing in 
any of the classes, whatever be the numbers of ternis in any classes ; and 

^4. If ths terms are all e<]aal, but arranged in classes with different numbers in 
each, the average is equal to each of the figures, whatever be the numbers of the 
figures in any of the classes. 

A meaning in this last proposition is that the statement is universally true, 
whatever be the weighting of the classes. In other words, vhen all the terms 
are equal, the weighting is indifierent. 

§ 5. What has bsen said of averaging any terms is true whether the terms 
be figures (as above described) or variations (of figures). For example, ap- 
plied to variations, the fourth proposition becomes : If the variations are all 
equal, the average is equal to each of them, no matter what be the weighting of the 
variations. 

Tliat the formuhe for the three averages satisfy this principle will be indi- 
cated in the course of the separate treatment of them. 

^ 6. Furthermore, if the variations are all equal, the variation of the aver- 
ages separately draivn of the figures which have varied and of the figures to which they 

have varied I . ^ ) '5 the same as the average of the variations (V), itself the same 

as any one of the variations I e. g. - J . 

If all the figures in each set are equal, this is self-evident. In other cases 
a demonstration may be needed for each kind of average. Indications of the 
demonstration will be added in the treatment of each kind of average. (See 
below, notes in II. i! G, III. ^ (5, and V. ^5). 

Of course, if the variations are all \, or 1, that is, if there are no varia- 
tions, this principle applies, and is now self-evident, whatever the figures. 
For in this case the corresponding figures in the two sets are identical, the 
sets themselves are so, and so must be their averages ; wherefore there is no 
variation of the averages. 

We shall be interested, in what follows, principally with sets of figures in 
at least one of which at least one of the figures, and one of the variations, is 
different from the rest — or rather, with general conditions which admit the 
possibility of such divergence. 

§ 7. Even now there is another case, common to all the kinds of averages, 
in which the variation of the averages and the average of the variations always 
agree. If all the figures at the first period are imits, wherefore the average also 
is a unit, the figures in the second set themselves express their variations from 

the first period, since in all the expressions for the variations, -^ , ^ , , 

the denominators are 1, and may be dropped. Therefore the expression for 

A. 

the variation of the averages, ^- , in which the denominator is also 1, and 

may be dropped, and in which the numerator is the average of the figures at 
the second period, is also the expression for the average of the variations. 
Thus 



500 APPENDIX A 

If all the figures at the first period are units, the variation of the averages and the 
average of the variations are alwai/s the same. 

This proposition, however, is very different from the preceding. In the 
conditions there posited it is indifferent what weighting is used. Here it is 
important that correct weighting be used, and it is necessary that the same 
weiglitingbe used in averaging the figures (of the second period) and in aver- 
aging the variations. 

Moreover this proposition is true also of all cases v)hen the figures at the first 
period are equal, whether they all be units or any other quantity. For this 
other quantity may be taken as a unit, and the figures in the second set be re- 
duced on the same scale, and then the proposition applies. 

Consequently this principle will form part of all our separate treatment of 
the three kinds of averages ( except in the third, where it will be swallowed 
up in a wider principle ) . The particular demonstrations Avill therefore be 
given later. (See 11. § 6, III. § 6, V. § 5). 

Applying to all cases, the following principles are also plain. 

§ 8. Given a set of figures (or variations) not all alike, of which the aver- 
age (of any kind) is known, if we add to it a figure equal to the average, or 
substract from it a figure which happens to be equal to the average, we do not 
alter the average. Hence it does not matter how often a figure equal to the 
average be added or subtracted, or whether it be omitted altogether. In other 
words, the iveighting of a figure equal to the average is indifferent. 

The similarity of this to § 4 is patent. 

§ 9. Given a set of figures (or variations) as before, if we add a figure larger 
than the average, we raise the average somewhat ; and if we subtract such a fig- 
ure we take away one of the influences that have made the average as high as 
it is, and so we lower the average. Hence it does matter how often such a 
figure is repeated, and to increase the number of times it is repeated or to en- 
large its iveight, is to raise the average nearer to it, and nearer and nearer the more 
we repeat it (but the average can never reach it, short of an infinite number 
of repetitions, for in that case its weight would be indifferent); and to de- 
crease the number of times it is repeated, or to diminish its iveight, is to loiver 
the average (until the figure is omitted altogether, when the average is what it 
would have been without this figure). 

1 10. Given a set of figures (or variations) as before, if we add a figure 
smaller than the average, we lower the average somewhat ; and if we subtract 
such a figure we take away one of the influences that have made the average 
as low as it is, and so we raise the average. Hence it does matter here, too, 
how often such a figure is repeated, and to increase the number of times it is 
repeated, or to enlarge its iveight, is to lower the average nearer to it, and nearer 
and nearer the more we repeat it (but without ever reaching it, as before); 
and to decrease the number of times it is repeated, or to diminish its weight, is 
to raise the average (until the figure is omitted, when the average is what it 
would have been without this figure).^ 

^ Of course, if we add or subtract a figure equal to one average aud unequal to 
another average, we alter that other average. Or if we add or subtract a figure 
unequal to this average, and so alter this average, we may perhaps not affect 
another average (to which this figure may happen to be equal). We are dealing 
with each kind of average carried throughout. 



VAUIATIONS AM) AVKItACJKS oOl 

These three principles can eiisily be demonstrated in the ease of each of tlie 
three kinds of averaj^es. But tiiey are too phiin to need demonstration. 

A resume of them is that wlieii a figure is tlie same as tlie average its 
weighting is indifferent ; otherwise its weigiiting eounts, and an alteration in 
the weighting of tlie classes without any variation in the sizes of the figures 
(but like a variation in the size of the figures) causes the average to change, 
the influence of an alteration in the weighting being different according as the 
figure operated on is larger or smaller than the average. To increase the 
repetitions of a larger figure has the same influence as to enlarge the figure ; 
and to increase tho repetitions of a smaller figure has the same influence as 
to diminish the figure ; and conversely. 

A practical application of these three principles is that an error in our 
weighting of a figure eipial to the average is of no account (i. e., if the aver- 
age without a figure is the same as the average with it, we need not concern 
ourselves about its weight). And the nearer a figure is to the average, the 
less an error in its weighting will count ; and the more it will count, the more 
the figure is removed from the average. Of coni'se in the case of an average 
of variations, what is here said applies to a variation according as it is the 
same as, near to, or far from, the average of the variations. 

Hence lastly, 

§ 11. Tf all the weights are altered in the same proportion, there is no effect upon 
the average, or, in other words, if we have the proper Aveights, we may alter 
them as we please, so long as we keep them in the same proportion (multi- 
plying or dividing them all by the same quantity). 

For if we increase all the weights in the same proportion, the increase of 
the weights of the figures equal to the average has no influence, and the in- 
crease of the weights of the figures larger than the average tends to raise the 
average, while the increase of the weights of the figures smaller than the 
average tends to lower it ; but as the influence of all the figures below the 
average to lower it is equal to the influence of all the figures above the aver- 
age to raise it (for otherwise the average would not be where it is), so the in- 
fluence of the changes in the numbers of the former is equal to the opposite 
influence of the proportionally equal changes in the numbers of tlie latter, 
and the average remains where it w^as. And reversely if we decrease all the 
weights in the same proportion. 

If this general explanation be not sufficient, the proposition may be dem- 
onstrated in the case of each of the averages separately. This will be done 
incidentally for two of them in the following pages. (See I. §7, II. § 7). 

Or another general proof may be made as follows. If, for instance, we 
double all the weights, we may segregate all the new terms, and so form two 
distinct sets of figures (or variations) exactly duplicating each other, one of 
them being the original set. Then the average of each set, separately drawn, 
will be the same. C'onse(iuently the one average of the two together will be 
the same. And similarly, whatever be the multiplier. 

Thus the general system of Aveighing depends upon the relative sizes of 
the weights, and not upon their absolute sizes. However large or small the 
weights, if in the same jiroportion, we practically have the same weighting. 

This being so, we have even weighting whether we count every figure in the 
sets ( or every variation ) only once, or an equal number of times. 



502 APPEXDIX A 

ir. ARITHMETIC AVERAGIXG. 
§ 1. With single figures in the two sets, the arithmetic average at the first 

periodis - (a, -|- 6, -f- to n terras), and at the second it is - (a2 + 62 4- 

n '^ 

to n terms) ; wherefore the variation of the averages is 

A, V'" + '" + ) 



which reduces to 



A, «2 + b.i \ 



Ai «i +6i + ' 

(thus showing, incidentally, that it is indifferent whether we compare the arith- 
metic averages or the sums). 

Here the weighting of the figures in each average is even, each figure count- 
ing once. 

^ 2. Given the same single figures, the arithmetic average of the varia- 
tions, likewise with even weighting, each variation counting once, is 



n \ «! 6i / 



This is a different expression, not univei-sally reducing to the preceding. 
§ 3. With classes of figures, the arithmetic average at the first period may 
be expressed in full thus, 

Ai = -, {(«iH- «i + tox terms) 4-(^i + *i+ to i/ terras) + 

to?i classes L 

which may be abbreviated to -, {m^ -f-y^i + to « terras), or raay also be 

written ' ^'^'^ '^ i + " " — _ ^j^^j ^\^q average at the second period is like 

■C + 1/ + 

unto it, with change only in the numbering of a , b, Therefore the 

variation of the averages is 

A -/ ( ■^'«2 + 2/^2 + to n terms ) 



^ -> (-^'f'l + y'*! + to n terms) 

which reduces to 

A2 m2-ryb2-\- 



Ai -i-Oi+yfti-l- 

(in which, again, the variation of the arithmetic averages is the same as the 
variation of the suras ) . 

Here the weighting of the figures in each average is directly according to 
the nurabers of times the figures are repeated in each class. Naturally an 
average of the figures taken only once each, as in § 1, is not an average of the 
figures here supposed, but only of the figures there supposed. ( Yet the evenly 
weighted average might be said to be the average of tlie classes simply as 
classes, each class counting as an individual, without regard to the numbers 
of figures in tliem) 



VAIMAI'IONS AM) AVKI{A(;ES 50;$ 

§4. (Jiven the siiiiu- classes of liuures, tlie similarly weij^lited arithmetic 
averaj^e of the variations in full is 

V i^ {(;!; + «;+ <-■ terms) + (';+';+ to , terms) -f ' 

to // classes J- , 

which mav be al)l)reviatetl to — ( ,i '' -f- '/ ," -\- to ii terms | , or mav be 

written 

"■.' , ''-2 I 

Y "i f-i 

.'■+.'/ + ■ 

Again this expression is difl'erent, not universally redueiui^ to the preeeding. 

§5. When dealing with single figures, we may say that we are dealing 
with classes in each of wliich there is only one individual. Therefore we may 
subsume sets of single figures (or variations) under sets of classes of figures 
(or variations), and treat only of the latter. Thus in both the preceding cases 
we have been dealing with arithmetic averages of classes in which the weiglit" 
ing is according to the numbers of figures in the classes, and of averages of 
variations with weighting likewise according to the numbers of varying figures 
in the classes. 

The conclusion from the preceding paragraphs then is that the variation of 
the arithmetic averages and the arithmetic arrra(/e of the variatiom, in all cases with 
weighting according to live numbers of figures in the classes, are not universaUy the, 
same. 

^6. Tlie variation of the arithmetic averages and the arithmetic average of the 
variations, all with weighting according to the numbers of figures in the classes, are 
the same when all the figures at the first period are eqiud. 

Let «j = 6^= =;s (s being ant/ figure above, 1)elow, or at unity, in- 
tegral or fractional). Then the expression for tlie comparison of the averages 

, , ,. , . . . -, , xa., -\- yh« 4- , , 

(that tor the sums) m i .-> reduces to —^ — r"^, c ; and the expression 

s(.i-4-2/+ ) 

for the average of the variations in § 4 reduces to the same. Q. E. D. 

Hence if we reduce all the actual figures in an irregular set at the first 
period to tlie same ideal figure (by taking some common divisor of them) 
reducing the figures at the second period in the same proportions, and if we 
change the numbers of the figures in the classes in the inverse proportions ( so 
as to keep unchanged the sizes of the classes), the weighting desired in order 
to make the two methods agree will be in accordance \<ith the adjusted num- 
bers of these ideal individuals in the classes. 

Thus a factor which affects the variation of the averages is the number of 
times the figures occur in each class, or weighting t)f the variations directly 
according to the numbers of the figures. 

Tn all other cases (except of course when all the variations are equal — 
see I. ? 6 M the average of the variations will re(iuire ditterent weighting of 
the variations to make it agree with the variation of the averages. 

1 In the formula above given in Ml«?t " ^ ,^ = = r, and the expression 

reduces to r. Now a2 = »'«i, y^ ^ >'bi, and so on. Introduce these into the for- 
mula in § 3, and this also reduces to /•. 



504 APPENDIX A 

I 7. When the numbers of the figures in the classes are all equal, the variation of 
the arithmetic averages {each with weighting according to the numbers of figures in the 
classes — in this case with even weighting) is the same as the arithmetic average of the 
variations vnth weighting according to the sizes of the figures at the first period. 

Let x = y^ = i. Then the expression for the variation of the aver- 
ages reduces to -~ 1-^-7- i > and this, by dropping the t from both sides 

of the fraction, to the expression for the variation of the figures taken singly, 
i. e., with even weighting. We want now to prove that this expression is the 
same as the expression for the average of the variations with weighting ac- 
cording to «i, 61, 

The expression for the average of the variations with weighting according 

to a-i, 6,, is -7, I Oi ' + ^1 ,^ + to n terms ) in which 7i'''^a, + 

n" \ «i Oi / 

6i -f- to )i terms. This expression, by reducing, and restoring the value 

of n^\ becomes — , ,^ , , which is the same as the expression for the 

«i + 6i+ 

variation of the averages. Q. E. D. 

Hence a factor which affects the variation of the averages is the size of the 
figures in the classes, or weighting of tlie variations directly according to the 
sizes of the figures. 

The reason for this it is important that we should perceive. Take the 
simple case of two figures in each set, or two reported variations. Suppose <Xj 
is larger than b^ by an integi'al number of times, say q. Now the single vari- 
ation of Oj to ffj) compared with the single variation of 6j to b.^, is a variation 
not merely of a g times greater quantity, but of q times more quantities. 
Hence it virtually contains q times more variations. Therefore, as, according 
to the preceding proposition, the average of classes of variations, when the 
figures at the first period are equal, must be drawn with the classes weighted 
according to the numbers of figures in them, if in the present case the weight 
1 be given to the figure 61, the weight q sliould be given to the figure Oj, as 

really being a class composed of q b^' s, each of which varies by -^ . In fact, 

the formula for the variation of the averages may be analysed into this, 

(I f'o ,,09, \ , 7 b., 
61 ~ + 0, ' -f- to & terms + 0, ,' 
1^1 ^1 / bi 

{bi-\-bi-\- to q terms ) + ^1 ' 

b. 



which reduces to 



' Va,^ bj ^ 1 / o., , 6.,\ 1 . , . J 
— , , , .. / — and to ^7 ( q '-\- ," I , winch is the expres- 
6,(5 + 1) n'' V^«i bj 

sion for the average of the variations with q and 1 for the weights. What 
is here shown of two figures so conveniently related may be generalized as fol- 
lows. Having one reported variation of Oj to a.^, one reported variation of 6, 
to 62) iind so on, we may view the variation of a^ to «., as consisting of % 

variations of 1 I = - | to " , and the variation of 6, to b., as consisting of 6, 
variations of 1 i = , ' j to .' , and so on with all the other reported single varia- 



to 1.^ in that it i-ontaiiis fifteen such variations ; for ^^ = ,^^ ' , and 



VARIATIONS AM) AVKIJACJKS 505 



tion.s. Then tin- weight of a, variations of ] to '"' is a,, and tiiat of/*, varia- 

"i 

tions of 1 to " is/;,, and similarly in all the other rases; and the rigorous 

expression for the arithmetic average of these variations, according to § o, 
is as above given. 

This way of viewing the variations is a perfectly correct way of viewing 
them, tliough not the only correct way. In a variation, for instance, of 15 
to 20, the variation element is a variation of 1 to I5, or a variation by |. 
But the nominally single variation of 15 to 20 differs from the variation of 1 

20 __ liX 15 
15 ~" 1X15 

have a variation, not of 1 by |, but of 15 by |. Now in a comparison of the 
arithmetic averages at the two periods (with weighting according to the num- 
bers of figures reported in the classes, their reported sizes also being used) this 
difference between the variation of 15 to 20 and the variation of 1 to IJ — 
these figures being supposed to appear in two otherwise similar sets — -shows 
itself by the fact that a different result is obtained according as we use the one 
or the other of these sets, although they contain the same variation elements. 
But in tlie arithmetic average of the variations ( with weighting likewise ac- 
cording to the numbers of figures and variations reported in the classes) this 
difference is not allowed for, tlie variation of 15 to 20 and the variation of 1 
to 1 J having exactly the same influence upon the result. 

The former may, then, be the truer expression even for the average of the 
variations — and the average of the variations must be adapted to it by using 
a difTerent weighting^in all those problems in which we wish the variation 
to count according to the sizes of the figures ; ^ but not otherwise. 



-' In general we want the size factor in the weighting of variations in all those 
cases in which greater effort is needed to produce the same variation in a greater 
than in a smaller quantity. Also the average of the variations with weighting 
according to tlie numbers actually reported in the classes can obviously be correct 
only in those problems in w hich it is possible to state the figures only in one way ; 
for otherwise the average of the variations would depend upon the accidental way 
in which the figures that vary happen to be reported. It may happen even that 
we wish the variations to count inversely according to the sizes of the figures. 

Then the weiglits of the variations would be , , , , and the formula, 

-^2 , _^ , 



1 1 

- + r + 

«1 0] 

Or if in any eases the weighting should be inversely according to the sizes of the 
classes, viz., — ,^r-i , the formula would be 

O2 bo 

— -^ — I- — -^ — f- 



1 1 

+ --7 + 

■'•«i yb^ 



506 APPENDIX A 

We have discovered two factors in the weighting of the variations — the 
numbers of the figures employed, and their sizes at the first period. We 
have discovered each of these upon eliminating the other. We must now 
unite the two. As they both act directly, their influences work together and 
strengthen each other. Therefore, in all cases, 

§ 8. The variation of the arithmetic averages of classes of figure^ each of these 
averages with weighting according to the numbers of figures in the classes, is the same 
as the arithmetic average of the variations with weighting according to the sizes of the 
classes at the first period. 

The expression for the average of the variations with weighting according 

^2 I 7 ^2 1 

m rVOiT-+ 17 1 

7 . '^a, "^ ^6, 1- 1 J . xa2-\-yo2 + 

to xa^,^lo,, , is ^ — j — f- , which reduces to . , , , 

■^ ' xa^'\-ybi+ ' xa^ + yb,+ ' 

which is the expression for the variation of the averages. Q. E. D. 

In any such expression as the last we should notice that the mathematical 

terms are not x and y, , a^ and 6^, , but xa^, yb-^, Therefore 

in the denominator the one term xa^ may be replaced by ai, the one term yb^ 

by bi, and so on. Then iu the numerator the terms become aj — , bi t^ > and so 
on. The expression then falls under § 7. 

It may be remarked that the method of averaging the variations has 
this advantage, that it tells us what we are doing, while the method of com- 
paring the averages hides this. We have now discovered that when Ave com- 
pare the arithmetic averages of the figures which have varied and of the 
figures to which these have varied, we are virtually averaging the variations 
with weighting according to the total sizes of the classes at the first period. 
Hence we may view this weighting of the variations as hidden or latent in 
the method of comparing the averages. 

From this follows a simple corollary : 

^^. If the sizes of the classes at the first period are all eqiud ( i. e. , 

xttj =: yby = ), the variation of the arithmetic averages, each with weighting clc- 

cording to the numbers of figures in the classes, is the same as the arithmetic average 
of the variations ivith even weighting. 

This condition is brought about when the numbers of the figures in the 
classes are in inverse proportion to the sizes of the figures at the first period. 

III. HARMONIC AVERAGING. 
? 1. With single figures in the two sets, the harmonic average at the first 
period is :p— j z. r ; and that at the second is like unto it. There- 



lfA-ul + V 

n Voi ' 6i / 

fore the variation of the averages is 



ifi+l+ ^ 

A2 __ w V g^ b^ / 

Ai~ 1 

n ' a, ^ ft, ^ ) 



VARIATIONS ANI> AVKUAUES 507 

wliic'li reduces to 

V +'+ ] 

A. 171+1 + .:: V 

and to 

1 + 1+ 

Ai 1 + 1 + ' 

(which sliows that it iw indifferent, in comparing liarnionic averages, whether 
we inversely compare tlie arithmetic averages of the reciprocals of the figures 
or the sums of the reciprocals of the figures). 

Here the weighting of the figures in eaeli ;iverage is even, eacli figure 
counting once. 

§ 2. Given the same single figures, tlie liarnionic average of the varia- 
tions, likewise with even weighting, each variation counting once, is 

1 



V 



" v "■; t ) 



wliich reduces to 

v= ' 

if«i+^i + ^ 

n \ a., ^ b,^ J 

This is a different expression, not universally reducing to the preceding. 
§3. With classes of figures, the liarnionic average at the first period may 
be expressed in full thus, 

to n classes 

which maybe abbreviated to — . ■ — , or may also 

, ( + ; + to n terms ) 

be written —^ -^"^ . And the harmonic average at the second period 

is like unto it. Therefore tlie variation of the averages, after reductions 
similar to those used in § 1, is 

■': + i + 

^^ ■^+f + ' 

Here, too, the weighting of the figures in each average is directly accord- 
ing to the numbers of times the figures are repeated in each class. No other 
way of averaging tiie sets would average the figures supposed. 



508 



APPENDIX A 



^ 4. Given the same classes of figures, the similarly weighted harmonic 
average of the variations in full is, after the first reductions in the denomi- 
nators, 

V = 1 

M(°-;+«-:+ '°="'--)+(t;+t:+ to!,ter..)+' 



to n classes 



which may be abbreviated to 



} 



"/I 3^ + V i + to m terms I 



or may also be written 



^-\-y + 



'a,-^n,'-^ 

Again this is a different expression, not universally reducing to the pre- 
ceding. 

^ 5. Here also we may subsume single figures under classes, and confine 
our attention to the latter. Thus, 

The vanation of the Jmrmonic averages and the hfvrmonic average of the valua- 
tions, in all cases with weighting according to the numbers of figures in the classes, are 
not universally the same. 

I 6. The variation of the harmonic averages and the harmonic average of the 
variations, all with iveighting according to the numbers of figures in the classes, are 
the same when the figures at the first period are equal. 

Let a^ =: 6j = = s. Then the expression for the comparison of the 



averages in I 3 reduces to 



! + !/ + 



(- 

\a. 



+ 1 + 



; and the expression for the 



average of the variations in ^ 4 reduces to the same. Q. E. D. 

Here is the same factor affecting the variation of the harmonic averages as 
we found in the case of the variation of the arithmetic averages, namely the 
number of times the figures are repeated in each class, or weighting of the 
variations directly according to the numbers of the figures. 

In all other cases (except of course when all the variations are equal — 
see I. ?6i) the average of the variations will require dift'erent weighting 
of the variations to make it agree with the variations of the averages. 

§ 7. When the numbers of the figures in the classes are all equal, the variation of 
the harmonic averages {each with weighting according to the numbers of figures in the 
classes — in this case with even weighting) is the same as the harmonic average of the 
variation.'^ ivith tveighting inversely according to the sizes of the figures at the first 
period. 

Then the expression for the variation of the aver- 



Let .r 



// = 



1 Here also let -" 



62 



r , and introduce r in the formula given here 



in ^ 4, and it reduces to r . Then aj = ra^^ , b^ = rb^ , and so on ; and by intro- 
ducing these values in the formula in § .3, it, too, reduces to r . 



VAKIATIOISS AND AVEHA(;KS 509 

'(,;+;+ ) 

ages in § 3 reduces to ' ' , ami this, by flr()pi)ing tlie t from 



'(:+^ ) 



l)oth sides of the fraetion, to the expression for the variation of the averages 

of the figures taken singly, i. e., with even weighting. We want now to 

prove that this expression is the same as the expression for tlie average of the 

. , . , . ,. 11 
variations witli weighting according to , , 

The expression for the average of the variations so weighted is 

=— ^ , , , X , i" which »/^= 4- , + to /( 

—A • '+ , • ,' + to « terms ' • 

" \ "l '1 2 "l "2 / 

terms ; wherefore tliis expression, hv reducinir, restoring tiie value of //'. and 

,;+,;+ ■ 

converting, becomes ' ' , which is tlie same as the j>rereding. 

+ ,+ 

Q. E. D. 

Thus here, too, a factor which affects the variation of the averages is the 
size of the figures in the classes ; only here the weighting of the variations is 
inversely according to the sizes of the figures. 

This is the opposite of what Avas the case in the variation of tiie arith- 
metic averages. There the larger a figure, the more its variation counts. 
Here the larger a figure, the less its variation counts. 

Of course in the harmonic average, as in all averages, the larger a figure, 
the larger is the average of the set of figures in which this figure occurs ; 
and similarly, the larger a variation, the larger is the average of the set of 
variations in which this variation occurs. The proposition before us is that 
the larger tiie figure which varies, the smaller is the influence of its varia- 
tion upon the variation of the averages. The reason of this is because the 
harmonic average is the reciprocal of the arithmetic average of the reciji- 
rocals ; but the larger a figure, the smaller is its reciprocjil. 

We have, then, discovered two factors in the weighting of the variations 
— the numbers of figures, and their sizes at the first period. But as the one 
of these acts directly and the other inversely, their combined influence is the 
balance left over as the one neutralizes the other. Therefore, taking both 
into account, in all cases, 

§ 8. The variation of the harmonic averages of classes of fic/ures, each of these 
averages with loeighting directly according to the numbers of the figures in the classes, 
is the same as the harmonic average of the variatiotis with iveights which are the ratios 
of the numbers of figures in the classes to the sizes of these figures at the first period. 

The expression for the average of tlie variations with weigliting according 

•'-f ^+ ■'+•' + 

X y . a, &i ,-11 ^'i ^1 

to , , , IS - ^—j , whicli reduces to , 

«! Oi ■'■."' _l_ y . *i -U ■' ^. •/ _L 

«, a^ h^ bn a.. b.^ 

which is the expression for the variation of the averages. Q. E. D. 
From this also follows the simple corollary : 



510 APPENDIX A 

§ 9. If tlie ratios of the nwnbers of figures in the classes to the sizes of the figures at 

the first pei'iod are all equal I ;'. e. — = ^ = ] , the variation of the harmonic 

averages, each with weighting according to the numbers of figures in the cktsses, is the 
same as the harmonic average of the variations with even iveighting. 

This condition is brought about when the numbers of the figures in the 
classes are in direct proportion to the sizes of the figures at the first period. 

§ 10. A remark deserves to be added. 

We have seen that in the arithmetic averaging the expression for the varia- 
tion of the averages was in some cases truer than the expression for the aver- 
age of the variations (unless this has its weighting specially adapted), because 
it gives weight to the figures according to their numbers and sizes in the de- 
nominator. NoAV in the same case we might here still want the figures to 
count directly according to their sizes as well as directly according to their 
numbers^— we might Avant the variation of a larger number to count more in- 
stead of less. Therefore the comparison of the harmonic averages would not 
be serviceable for this purpose, and in order to carry it out we must insert the 
desired weighting in the expression for the harmonic average of the varia- 
tions, thus, 

1 
V 



which reduces to 



—7, 1 .10, — -^ i/0, , + to ?i terms 

?i" \ rt., Oo / 



Y 



I h 2 



or, if we represent xa^ by a, yb^ by b, and so on, (or supposing the weights 
are sometimes wanted to be something else, still letting a and b repre- 
sent them) we should have 

1 



-77(a — -|-b7^ + to n terms ) 



in which j/'' = a + b + to n terms. The expression for the variation 

of the averages, however, is serviceable on condition that the figures at the 
first period are reduced to equality ; for then the numbers of times they are 
repeated in their classes are proportional to a, b, 

TV. CASES OF AGREEMENT BETWEEN THE ARITHMETIC AND 
THE HARMONIC AVERAGES OF VARIATIONS. 

We have noticed in the comparison of the arithmetic averages what hap- 
pens when Oj = 5i= ,and when xa-^ ^ybi = , and in the comparison 

of the harmonic averages what happens also when Oj = 6j= , and when 

— ^ ^= There remains to see what happens, in the former case, when 

«i h 

0,^62=' y ai^d when xa^^^yb^= , and in the latter, when 

X y 
a., = J^ ^ and when — = f-= A few other coincidences will 

also call for attention. 



A^AHIATIONS AND AVEllAGPiS 511 

§ 1. Tlie variation of the arithmetic averages being expressed thus, 

A2 ^ xa^-j-yh^_-\- ■■•■•■ 
Ai jai + ?/6,+ ' 

wo know this to he the same as the arithemetic average of tlie variations with 
weighting according to roj, yh-^, 

Leta2 = 62= = s. Then /r/, = /ctj ' ^ •' '« ', imd siniihirly ybi = 

"2 '^'2 

)/•<,', and so on. Therefore this exiiression becomes ', , 

«2 h., 

which reduces to ' , , which we recognize as the expression 

%:+.":.:+ /_ 

for the liarmonic average of the variations witli weighting according to 

X, y, Therefore, 

If it happens that the figures at the second period are all equal, the arithmetic 
average of the variations with tveighting according to the sizes of the classes at the first 
period is the same as the harmonic average of the variations ivith ireighting according 
to the numbers of figures in the classes. 

§ 2. Let .raj =^ yb2= =^t- Then «(,=::x«2 ^ = ' ' , and similarly ybf 

a^ f(2 

t c , and so on. Therefore the above expression becomes , 

"2 *2 

which reduces to -l — -^ j . , which we recognize as the expression 



n V02 O2 / 



for the harmonic average of the variations with even weighting. Therefore, 
If it happens that the sizes of the classes at the second period are all equal, the 

arithmetic average of the variations with weighting according to the sizes of the classes 

at the first period is the same as the harmonic average of the variations with even 

weighting. 

This theorem may be extended, as follows : 

'i 3. The harmonic average of the variations with weighting according to 

X -{- V -\- 

the numbers of figures in the classes being this, " , , suppose 

\.^'b.+ _ 

tliat instead of this weighting we use weigiiting according to the sizes of the 

chisses at the second period. Then we must substitute joj for .r, yb.^ for y, and 

, , .!•«, -(- w6, 4- , . , 1 ••"•2 + '/^2 4- 

so cm, and we have — "^ — r , winch reduces to , , — , , 

"2 ''2 
whicii we recognize as the exjjression for the variation of the aritmetic aver- 
ages with weighting according to .r, y, , and which we know to be the same 

as the arithmetic average of the variations with weighting accoi'ding to 

xa^, ybi Therefore, in all cases. 

The arit-kmetic average of any vai-iations with weighting according to the sizes of the 
classes at the first period is tlie same as the hai~monic average of the variations vnih 
weighting according to the sizoi of the classes at the second period. 



512 APPENDIX A 

§ 4. The variation of the harmonic averages being expressed thus, 

f-l-^4- 

^"'^ + ^4- ' 

M-e know this to be the same as the harmonic average of the variations with 
weighting according to the ratios — , ~- , 

Let r(,, = 6, ^ =: s. Then o, = s — and — = -^ , and similarlv ," = 

02 a, sdj • Oj 



-^ , and so on. Therefore this expression becomes ^ — , wliich 

soi i^ 1 !_)_ 

s s 

reduces to — ' — , 7 , wliich we recognize as the expression for the 

■••4-v4- 

arithmetic average of the variations M'ith Aveighting according to r, y, _. 

Therefore, 

If it happens that tJie figures at the second period are all equal, the harmonic aver- 
age of the variations loith weighting according lo the ratios of the numbers of figures in 
the classes to the sizes of the figure.? at the first period is the same as the arithmetic 
average of the variations ivith iveighting according to the numbers of figures in the 
classes. 

o - r •'■ y mi 1 •'^ ''"•' 1 • • 

§0. L/Ct =:~^ ^r. ihen a = /■((,, and — ^ — - ; and sinu- 

a, 62 a-i a^ 

larly , ^ ,— ! and so on. Therefore the above expression becomes 

r H '■ zT + -, / T X 

«i "1 , • 1 1 1 / o, , 0, , \ 1 • , 

— *— , f , which reduces to - ~" + 7^ + , which we recog- 

r + r+ ?iV«i 61 /' 

nize as the expression for the arithmetic average of the variations with even 
weighting. Therefore, 

Jf it happens that the ratios of the numbers of figures in the classes to the sizes of 
the figures at the second period are all equcd, the harmonic average of the varia- 
tions with weighting according to the ratios of these numbers to the sizes of the figures 
at the first period is the same as the arithmetic average of the variations with even 
weighting. 

This theorem also may be extended, as follows : 

§ 6. The arithmetic average of the variations with weighting according to 

the numbers of figures in the classes being this, — ^-r- — ~ , sup- 
pose that instead of this weighting we use weighting according to the ratios 
of the numbers of figures in the classes to the sizes of the figures at the sec- 
ond period. Then we must substitute - for .r , "- for 1/ , and so on, and we have 

a., 0,, 



VARIATIONS AND AVERAGES 513 

(«, a, 0^0, ,-11 "1 ^, ... 

- , whicli reduces to , which we recog- 

^+1 + - + ^4- 

nize as the expression for the variation of the harmonic averages with weight- 
ing according to x, y , , and which we know to be the same as the har- 
monic average of the variations with weighting according to > j- , • 

Therefore, in all cases, 

21ie harmonic avemc/e of the variations with weighting according to the ratios of 
the numbers of figures in the classes to the sizes of the figures at tlie first period is the 
same as the arithmetic average of the variations with weighting according to the ratios 
of these numhers to Hie sizes of the figures at the second period. 

^ 7. Suppose the numbers equal the variations. 

Then x^= ^ , 2/ = v^ , and so on ; and the harmonic average of the varia- 

lions with weighting according to x, w, becomes r^ — j , 

° '^ » .J> ^."ii^.^i I 

which reduces to ( ^ + .^ + ) , which we recognize as the arithmetic 

w \ a^ Oj / 

average of the variations with even weighting. Therefore, 

If it happens that the number of figures hi every class equals the variation, the 

arithmetic average of the variations with even weighting is the same as the harmonic 

average of the variations %vith weighting according to the numbers. 

I 8. Suppose the numbers equal the reciprocals of the variations. 

Then .r ^ - , y= J , and so on ; and the arithmetic average of the varia- 
f'2 "2 

^1.^2 J_^.*2 ■ 

tions with weighting according to X, 1/, becomes -^ — ' ~ , 

a, ^ b, ^ 
which reduces to ^r—z -. r- , which we recognize as the harmonic 



n \ 02 0.2 ) 



average of the variations with even weighting. Therefore, 

If it happens that the number of figures in every class equals the reciprocal of the 
variation, the harmonic average of the variations uith even weighting is the same as 
the arithmetic average of the variations with weighting according to the numbers. ^ 

^ Although aside from our subject, yet as throwing light upon it, the follow- 
ing two theorems may be added : 

The simple arithmetic average of any quantities is the harmonic average of 
them with iveighting directly according to their sizes. 

Of the quantities a, b, , k, the harmonic average with weights di- 
rectly according to their sizes is , which reduces to 

a-+6-+ + k-r 

a b k 

— (a -\- b -\- + k), which is their simple arithmetic average. Q. E. D. 

n 

33 



514 APPENDIX A 

V. GEOMETRIC AVERAGING. 
§ 1. With single figures in the two sets, the geometric average at the first 

period is i/^^Oj -bi to ?i terms, and that at the second is like unto it. 

Therefore the variation of the averages is 

A2_ i/^a2-^2 

Ai v^'^ai-^ ' 

( Here it may incidentally be remarked that the comparison of these averages 
cannot be replaced by comparison of the products except when the result is 
unity ; for the products are in a ratio the ?ith power of the ratio between the 
averages. ) 

Here the weighting of the figures in each average is even, each figure 
counting once. 

^ 2. Given the same single figures, the geometric average of the variations, 
likewise with even weighting, each variation counting once, is 

This is an expression which universally has the same value as the pre- 
ceding. 

§ 3. With classes of figures, the geometric average at the first period may 
be expressed in full thus, 

Ai = t/^ ( fh • «! to X terms ) • ( ^^ ■ 6^ toy terms ) ton classes, 

which may be abbreviated to i/^fli'' -b^^ to n terms. And, the average 

at the second period being similar, the variation of the averages is 

A2 v'^ o-i" • b^-' to w terms 

Ai i/'^«/ • V to '^ terms 

Here the weighting of the figures in each average is directly according to 
the numbers of times the figures occur in each class. No other way of aver- 
aging the sets would average the figures supposed. 

§ 4. Given the same classes of figures, the similarly weighted geometric 
average of the variations in full is 

The simple harmonic average of any quantities is the arithmetic average of 

them with xueighting inversely according to their sizes. 

Let k be the largest of these quantities. Then the weights of these quantities 

k k k 

according to the inverse of their sizes are — for a , — for b, , and -^ (or 1) 

for k . The arithmetic average of these quantities so weighted is 

■5 z ; I — ■ a + ^ ■ b + + k- y^ ) , 

k k k \ a b k '' 

a b k 
which reduces to — -, r which is their simple harmonic aver- 



1(^+7 + + 1) 

n ^ a b k I 

Q. E. D. 



VARIATIONS AND AVERAGES 515 

n' / ( a a., 



«'/ /f(, a., \ /bo b., \ 
= 1/ I • ' to .1- terms ) ■ I , • , " to */ terms ) 

to n classes, 



V 

w liicli may be abbreviated to 



V = i/(;!;y'-(';y ton term.. 

Again this is an expression which universally has the same value as the 
preceding. 

I 5. Here also subsuming single figures under classes, we see tiiat 

T'he variation of the (/eomelric averages and thec/eometric average of the variations, 
in all cases with weighting according to the numbers of figures in the classes, are the 
same. 

We see also that this agreement will universally take place with any 
weighting whatsoever, provided it be the same in all the three averagings. 

Thus, unlike the other two kinds of averaging, in the geometric averaging 
for this agreement to take place it is not necessary that rtj = ij :-= , or 

that ^ = ,- ^ ;^ since the agreement takes place not onlv in these but 

"i *i SI. 

in all cases. 

§ 6. If we employed here the weighting which we found necessary, except 

in the above two cases, for the arithmetic average of the variations in order 

that it should agree with the variation of the arithmetic averages, namely, 

according to .w,, v//j„ , which would be in this form, 



«"/ /a, X^"! f b.,\yh 

V \aj '\bj ■ ton terms, 

in wliich n'^ =^xa^ 4-^'! + to ?i terms ; or if we employed here the simi- 
larly necessary weighting in the harmonic averaging, namely—, 't- , , 

wliich would be in this form, 



/o„\"i fb.Xh 

\aj \bj ■ tonterms, 

in which »/■' = - + , - + to 7i terms ; either of these expressions for the 

"i ■ ^i 
average of tlie variations would agree with the expression for tlie variation 
of the averages (apart from the case when all the variations are alike) only 
in a particular case, namely if the figures are all the same at the first period 

(;'. e., fti = 6i= ) — the very condition previously found necessary to 

make averages of the variations with weighting like the weighting in the 
separate averages of the figures agree with the variations of these averages. 

^ That in this particular case the results are always the same as any one of the 
variations, whatever be the weighting employed (if only it be tlie same in the 
two averages of tlie figures that are compared), may easily be seen by letting 

-^ = T-= = = )•, and by introducing /• into the above expressions witli any 

weights. Tliese expressions then always reduce to r. 



516 APPENDIX A 

Therefore if here, with uneven variations, the figures being unequal at the 
first period, we reduce these to equality and adapt their numbers accordingly, 
and employ weighting according to the adapted numbers of figures in the classes, 
we shall get a result for the average of the variations thus weighted (really 
according to xa^, yb^, ) difierent from that of the variation of tlie aver- 
ages (in which the weighting is according to x, y, ). 

§ 7. The reason for the universal agreement in this case is that here the 
sizes of the figures at the first period, in the comparison of the averages, is 
offset by their sizes at the second period, and do not affect the result. Hence 
there is here no weighting in the comparison of the averages except the num- 
bers of times the figures occur in the classes. There is no hidden weighting. 

Therefore if we wish to employ weighting which also allows for the sizes 
of the figures at the first period (or at any other period), we have to intro- 
duce this into the separate averages which we compare as well as into the 
averages of the variations. 

VI. COMPARISON OF THE GEOMETRIC AVERAGE 
WITH THE OTHER TWO. 

§1. Of the same given numbers of the same given figures, at least one of 
which is unequal to the others, the arithmetic average is always greater than 
the harmonic. Thus 

xa-{-yb-\ -J x + y -Jr 



x + y+- -^ I ^ 1 

a^b^ 

The demonstration of this, which involves the demonstration that 
{xa + yb+ )(^ + f+ )>(-v + 2/ + Y, 

is somewhat elaborate, and need not be given here. 

This being true when the terms are figures (or integers), it is also true 
when the terms express the variations of figures (or are fractions). Hence 

With the same weighting, the arithmetic average of the same variations is always 
greater i or higher ) than the harmonic. That is, when the averages are above 
unity, the arithmetic average is a greater variation than the harmonic ; and 
when the averages are below unity, the arithmetic is a smaller variation 
than the harmonic ; or the arithmetic may be above while the harmonic is 
at unity, or may be at unity while the harmonic is below, or may be above 
while the harmonic is below. 

§ 2. Given only two figures (or two figures each repeated the same number 

of times, so that we have even weighting ) , the arithmetic mean being — - — , 

and the harmonic ., "^ _ ;= —^ — r , the geometric mean between these means is 
1.1 a -\- b' ° 

a"'"^ 

— jr — • — r-T ^ V ab , which is the geometric mean between the given fig- 
2 a-\-b ' ° CO 

ures. The same holds good if the figures are fractions (representing varia- 
tions of figures). Therefore 



v 



VARIATIONS AND AVERAGES 517 

Between two figures {or rurintions), each taken drnjly {or in equal numbers), the 
geometnc mean is the geometric mean between the arithmetic and the harmonic means. 
Hence also, in these cases, 

The geometric mean is sniaUer {lower) than the arithmetic, and greater {higher) 
than the harmonic. 

I 3. With three figures each taken singly (or in equal numbers), or with 

two figures taken uneven times (so tliat we must employ uneven weighting), 

the first of these propositions does not hold good of the averages between them. 

For between three given figures, each taken singly, the arithmetic aver- 

.a+6-t-c j„, • 3 Sabc , , 

age IS ^ , and the harmonic, -. - = -—-—- — _ • and the 

6 1 . X 1 ao -f-bc-\-cu 

a "*" b + c 



geometric mean between these two averages is -v/ w ^ 



S(obc 



3 ab -\- be -\- ca 

l./a + b+V^ r^-i-64-c" ,., J 

■\aoc\ , , , , 1 = I -, , 1 , which does not reduce to the geo- 

V \ab + bc-{- caj \ 1 , 1 i 1 

a b c 

metric average between the three figures, which is i/^a6c. Or with two 

figures of which the one occurs .c times, and the other y times, the arith- 

. xa-\-yb 1 ^, 1 . 2-4-V {x4-y)ab , 

raetic average is ?- — , and the harmonic, ^=- —-, ; and 

x + y ■'■ _|_ !' y" "I" •' -* 

a "'" b 



.1 . • 1 1 • xa -\- yb {x + y)ab 

the geometric mean between these two averages is -\ ~-^^ ■ ^^S- = 

' ^ X -\- y ya -\- xb 

I , / xa-\-yb\ \ xa-\-yb , . , , , , 

-V ab I , — r I = ^~, which does not reduce to the geometric aver- 

^ \ya-\-xb J ^ X y ' 

a b 
age between tiie two uneven classes, which is ' y a^bv. In general, a set of 
uneven classes of figures or variations may be analyzed into ^ larger number 
of even classes, some being homonymous. We may, therefore, confine 
our attention to sets of even classes, or of single figures or variations. 
Of a number, n, of such figures, the geometric mean between the arith- 

metic and the harmonic means is 



i 



a^ b ^ c 



— , which does not reduce to the geometric average of 



%i + !+U 

a b c 

the given figures, which is {/^ abc , (provided the figures, or variations, 

do not constitute, or reduce to, two classes with even weighting). 

But in these cases the second of the above propositions still holds, and we 
may also specify it more definitely, though without exact demonstration. 
Thus we have the two following propositions, the first of which is demon- 
strable. 

I 4. Of any figures, or vari(dions, with any weighting, the same in all the aver- 
agings, the geometric average is smaller {lower) than the arithmetic, and greater 
{higher) than the harmonic. 

The demonstration of the first part of this proposition, at least for cases 



518 APPENDIX A 

with even weighting (whence it can easily be extended) is generally given in 
works on algebra, • although notice of the second is generally neglected. Of 
both a brief indication is given by Walras (B. 61, p. 15). In full the demon- 
stration is too long to give here. 

^ 5. Of any figures, or variations, with any weightinc/, the same in all the averag- 
ings {provided it be not even weighting for only two classes of distinct figures or vari- 
ations), the geometric average is sometimes above, and sometimes heloiu, the geometric 
mean between their arithmetic and harmonic averages. 

This is shown by trial. But it is evident that, if different at all, the geo- 
metric average must vary on both sides of the geometric mean in question ; for 
if in any one set of figures or variations it be above, it must be below in a set 
of their reciprocals (e. g., it is above in 2, 3, 4, 5, and below in |, \, \, \). 

The easiest way to make the comparison is as follows. Place the expressions 
for this average and for this mean side by side, the former on the left and the 
latter on the right, thus, 



"■^a6c 



V 



square each, and raise each the n^^ power, thus, 
abc 



a e 
/ g-f &-f- c4- \" 



multiply each by the denominator of the one on the right, thus, 

{ahc Y (l^J^ljr\ + )", {a + b^c + )». 

The side on which superiority appears in the last line will also be the side on 
which it exists in the first line. But this method does not exhibit the pro- 
portionate amount of the difference. 

Trial seems to show that when all the terms are above unity, or when the 
terms in the numerators (including those virtually divided by unity) sum up 
to a greater amount than, or outweigh, the terms in the denominators ( in- 
cluding the omitted imits), the geometric average is above the geometric 
mean in question (as in \, 3, |^, 5); and if reversely, below (as in 2, \, 4, ^). 

In intermediate cases the geometric average may coincide with the geo- 
metric mean between the other averages. 

Trial also shows that the difference is generally very slight, although it 
may be appreciable if only a few of the figures or variations are very differ- 
ent from the rest, or very excessive compared with the average. 

I 6. Now of given numbers of given figures that vary between two periods, 
we have seen that the arithmetic average of their variations with weighting 
according to the sizes at the first period is 

^•^^; + ^^^^+ ^ xa,-^yb,+ . 

■^■«i + 2/&i+ ■!•% -t- 2/^1 -I- ' 

and the harmonic average of their variations with weighting according to the 
sizes of the classes at the second period is 

^ E. g. Todhunter's Algebra for the use of schools and colleges, § 680. 



VARIATIONS AND AVERAGES 519 



xa.^ + 2//>2 + _ J(i> + yh-i -•- 



:,a^«i + y6*i x-«i + A + 



We now see (from § 4) that the geometric average of these variations with 
weighting according to the sizes of the cbxsses at the first period is smaller 
than the arithmetic average with tliis weighting, thus, 



■■■/ (<^2Y\(b,Y^ . 



and that the geometric average of them with weighting according to the sizes 
of the classes at the second period is larger than the harmonic average with 
this weighting, thus, 

xa^+ybn /- 



vCT-iX) 



y''-. xa2 + yb^ + 



a-oi + yh + 



And we see (from § 5) that, generally, tlie geometric average with the latter 
weighting is larger nearly in tlie same proportion as with the former weight- 
ing it is smaller, than this one and the same average. Therefore if we take 
the geometric mean between tlie weights in these two systems of weighting, 
and employ the geometric average of the variations with weighting according 
to these means, thus, 



+ • 



1/ (:;) '"'-(t:) 



l^b^b^ 



the result will be nearly the same as that of the other average. But because, 
in § 5, the geometric average is not exactly at the geometric mean between the 
other two, but inclines to the one side or to the other according as the classes 
preponderate that rise or fall, so the geometric average with this intermediate 
weighting will Incline to the one side or to the other in similar cases. 

With only two classes with even weighting the geometric average is exactly 
at the geometric mean between the other two, as seen in § 2. Hence in these 
cases we should expect the following proposition to be demonstrable, and, in 
fact, find it so. 

§ 7. 0/" the variations of two figures, or classes of figures, such that the products 
{and hence the square roots of these products) of their sizes at both periods are equal, 
the geometric average with even weighting is the same as the arithmetic average with 
weighting according to the sizes at the first period, or the Imrmonic average with 
weighting according to the sizes at the second period. 

Treating of classes, as the more complex case including the other, we wish 
to prove that 

ft., _^ xa., 4- ybo 



4: 



*i -'"i 4- ybi ' 

given that xa^ ■ xa, = yb^ ■ yb., (this condition being necessary in order that the 
weighting of the geometric average may be even). 

From the condition we obtain Oj = ~^ ] and by substituting this value 

of Oj in the equation to be proved, we have 



520 APPENDIX A 



V 



yb-J}^ 62 _ ^<h 



+y\ 



which reduces to 



&i a-«i + yh 



which is evident ; wherefore the first equation is correct. Q. E. D. 

With only two figures, tliat is, with one figure in each class, x and y in the 
above are units (so that the condition is a^a2 = b-fi^), and the result works out 
the same. 

§ 8. With three or more figures, or classes of figures, even though the prod- 
ucts of their sizes at both periods be all equal, the statement in the preceding 
proposition does not universally hold. 

It holds when, there being no variations of the figures in the other classes, 
the figures in two such classes, or in any pairs of such classes (up to all the 
classes, provided there is no odd one that varies), vary to opposite geometric 
extremes so that the geometric average is unity ; for then the arithmetic aver- 
age with the weighting of the first period (or the harmonic with that of the 
second) is also unity, being so for each of the pairs and for the unchanged 
figures. 

It holds also in another case, which may be shown as follows for three 
classes. We wish to prove that under certain conditions 



Y'h.h.'^A — ^"2 + yh + gC2 
'^ Oi b^ c^ xor^ -+ ybi -}- zci ' 

a given condition being that xa^ • xa^ = yb-^ ■ yb^ = zc^ • zc^. From this condi- 

tion we obtain aj = "--^ and 63 ^ "irr • Upon substitution of these values 
X % y b^ 

in the equation to be proved, it reduces to 

1 ybi ■ zc^ -}- xa^ • zc^ -j- xai • yb^ 



(ccai)2- {yb-^Y- {zc^y xa^- yb-^^- zc^{xa^ + yby-\- zc^) 
This we see to be true when xa^ = yb^ = z<\ ; for then 



3 / 1 ^ ^{xa^Y 
(xoj)^ {xa-^yZxa^' 

which is evident. But then, according to the condition, we must also have 

xa^ = 2/&2 ^ 2^2, and both these conditions together mean that — ^ ^^ = — , 

Oi 0^ &[ 

that is, that the variations must all be alike. 

The same result is obtained if we analyze the case with four figures or 
classes, again with five, and so on. 

We know, moreover, the correctness of this result, because we know in 
general that when variations are all alike any average of them with any 
weighting is the same as the common variation. 

§ 9. In other cases trial shows that the geometric average of the variations 
with weighting according to the geometric means between the sizes of the 
classes at each period does not, in ordinary cases, much differ from the arith- 



VARIATIONS AND avp:iia(;es 521 

metic average of them with weighting according to the sizes of the classes at 
the first period or (which is the same thing) the harmonic average with 
weighting according to the sizes at the second period (there being a common 
form to which both of these averages so weighted reduce). 

More specifically, trial seems to show that the geometric average is above 
the other common form when the preponderating classes have variations rising 
above the average, and below the other common form when the prepondera- 
ting classes have variations falling below the average.'' Here the preponder- 
ance is to be determined by comparing the sums of the weights (measured as 
above described) of the former with those of the latter classes. 

The divergence of the geometric average from the common form seems to 
be greatest either when, amidst ordinary variations, the variations of the 
classes whose combined weights are about one fifth of the whole weighting 
are nearly the same, or at least in the same direction, while the rest are in 
the opposite direction, or when the variations of the smaller classes in the 
same direction are excessive. In all other cases the results are very close 
together. Especially so if none of the variations be great or unusual, or if 
great and extraordinary variations of some large classes are met by great op- 
posite variations of other nearly equally large classes, or of many small ones 
in combination nearly equalling them, or if the great variations of small 
classes are met by great opposite variations of nearly equally small classes. 
In general, if the combined weights of the classes rising above the average 
and the combined weights of the classes falling below the average are nearly 
even, these conditions tend to bring the results together ; and it is possible 
that they may coincide. 

VII. COMPARISON OF AVERAGES OF UNEQUAL SETS. 

For completeness we may add consideration of another case. So far we 
have all along supposed that the numbers of figures in all the classes separately 
and together are the same at both periods ; for only in this case can all the 
figures at the second period be regarded as variations of figures at the first. 
But it may happen that we can have different numbers of figures in the whole 
sets, or even a different number of classes ( since, so long as we are having a 
different number of figures, it may not matter whether the new ones be in old 
classes or whether they be altogether nevf ones forming new classes) ; and we 
may still have reason for wishing to compare the averages of these sets of 
figures. 

This case can be represented by numbering all the symbols previously left 
undistinguished. The numbers of classes of figures at the two periods may 
be represented by % and Ji, respectively ; the numbers of times the figures 

o^, 61, occur at the first period, by a-j, 2/1, , and the numbers of times 

the figures a^, b,,, occur at the second, by x.^, //q, ; and the total 

numbers of the individuals at the first period by n^^, so that n/ = i\ + ^i + 

to «! terms, and at the second by n./, so that «/ = x^ 4-^2 4" to 

n^ terms. 

2 If the geometric average shows a rise from 1 to 2, the rise of a figure from 1 
to IJ is virtually a foil compared with the variations of all the figures ; while if 
the geometric average shows a fall from 1 to J, the fall of a figure from 1 to | is 
virtually a rise compared with the variations of all the figures. 



522 APPENDIX A 

§ 1. Arithmetic averaging. The average at the first period is 
XjCij + y-fii -\- to «! terms 1 



, ^. T (^^itti + Vi&i 4- to Til terms); 

■h+yi + tojia^terms ' ri^^ ^ ^ ^ ' ^^ ^ ' ^ -" 

and at the second it is similar, with all the numbers changed. Therefore the 
variation of the averages is 

A —^ (•^■2«2 + 2/2^2 + ••'•••• to 7i2 terms) 

ii.2 ?l. 



A ~ 1 ' 

^ — 7(^i% + 2/A4- to rii terms) 

which reduces to 

A2 n/{x2a2 + 2/2^2 4" to fij terms) 

Ai ?i2^ (a;i% + 1/161 4- to «i terms ) * 

( Here, incidentally, we see that the variation of the averages is not the vari- 
ation of the sums, and can not be replaced by that, since the variation of the 
sums does not allow here for the change in the numbers of the figures. ) 

By restoring the values of n/ and n/, and rearranging, the expression may 
be stated in full, as follows : 

A2 X2(i'2 + 2/2^2 4" to n, terms a;i 4" 3/i 4" to ni terms 

Ai ^CiOi 4- 2/1^1 4" to % terms 3^2 4" 2/2 4" to 712 terms 

§ 2. Harmonic averaging. This being worked out in the same way, the vari- 
ation of the averages reduces to 

. — 4-|^4- to TCi terms . , , , 

A2 % Oj Xj 4- 2/2 "l~ to «.2 terms 

Ai Xo , ^Jo , , , 2:1 4- 2/1 4- to ?ii terms ' 

^ ~+r+ to Wo terms 1 ' ^1 ' 1 

§ 3. Oeometrie averaging. The variation of the averages obviously is 

A2 "^ "2^^ " ^2^^ to W2 terms 

■^1 "t/'^O'i'^i • ^i^/i to Til terms 

§ 4. All three of these variations of averages have the following common 
properties. 

Without any variation in the sizes of the figures between the first and 
the second period, a variation in the numbers of the figures in the classes — a 
change of weighting — may produce a variation in the average. These cases 
come under the principles in I. §§8, 9, 10. Thus, without any variation in 
the sizes of the figures, an increase or decrease in the number of any figure 
above the average at the first period will raise or lower the average at the 
second period ; and reversely an increase or decrease in the number of any 
figure below the average at the first period will lower or raise the average at 
the second period ; while any change whatever in the number of a figure 
equal to the average at the first period has no influence upon the average at 
the second period. 

Also, there being some variation in the average at the second period from 
the first, due to variations, all alike or otherwise, in the sizes of the figures 
(supposing no change in their numbers), then a superadded change in the 
numbers of the figures may accentuate, lessen, nullify, or outbalance the 
variation of the average due to the other influences alone. 



VARIATIONS AXD AVERAGES 523 

? 5. Naturally, if the numbers have not changed at all from the first to the 
second period, these formula; all reduce to the corresi)onding fornuihe with the 
same numbers at both periods. 

They do so also if the immhers of Ji(/)i.res in (he classes all van/ in the same pro- 
portion, and the number of the classes remains the same (('. e. rtj^Wi). We 
have then the state of things described in 1. §11 — retention of the same 
weighting ; and so, for instance, impossibility of a -variation in the average 
without any variation in the sizes of the figures. In effect, suppose from the 
first to the second period every dumber increases or decreases in the pro- 
portion r; then j'2 = rxj, 2/2=:?-?/,, and so on, and the expression for the 
arithmetic average becomes 

A2 ri\a2 -\- ry^b2 -\- to n terms Xy-{- y^ -{- to n terms 

Ai iYh + ?/i^i + to V terms rr^ + r?/, + to n terms' 

which reduces to 

A2 Xid-i "l~ 2/1^2 "l~ to ?i terms 

Ai x^a^ -}- 2/1&) + to ?i terms ' 

that is, to what is sometimes the proper form of the average of the variations 
on the numbers which are common to l)otli the periods — in this case the 
numbers at the first period. Or if we let r^ be the reciprocal of r, then 
x^ ■= r^X2 , 2/1 = r^y2 , and so on, and the expression will still reduce to what 
is in the same cases the proper form of the average of the variations on the 
numbers which are common to both the periods — this time, the numbers at 
the second period, wherefore it is indifferent whether we use the numbers of 
the first or of the second period. And similar would be the reductions in the 
expressions for the variations of the harmonic and the geometric averages, 
except that these would yield forms for the averages of the variations not 
proper when the above forms of the arithmetic averages are proper, the 
weighting of the variations being difTerent. 

^ 6. In all other cases there is no reduction of these formula; to any for- 
mulse for averages of variations. It is impossible to find any formula for any 
averages of variations which agree with any of the above formulae for the 
variations of averages. The reason is simple. There is no average of vari- 
ations in these cases, because, although there may be some variations con- 
tained in the two sets of figures, yet there are other figures whicli, appealing 
only in the one or in the other set, do not represent a variation of anything. 

The forms here reviewed are averages of the figures as reported — in the 
numbers that happen to be reported. In some matters it is necessary to alter 
these numbers into other numbers, in various ways, always according to some 
established principle. After such reduction, the new numbers being used in 
place of the old, the formulae will remain the same in form, although, being 
applied to different numbers, their results will be different. But it is some- 
times possible to introduce into the formulse themselves the principle by 
which the reduction of the numbers is made. One condition for this is 
that the new formuhi? may be made to contain something which necessarily 
confines them to the same number of classes at each period. Then the for- 
mulse themselves become difTerent in form. And now the new formulae may 
even have some points of contact with the formulae for averages of variation. 



APPENDIX B, 



ON COMPENSATOKY VARIATIONS. 

The supposition is that we have two series of three terms, a, m, b, which 
are such that a^m = b at one period and at another a and b are opposite 
terms, compensating for each other, in one of the three progressions, arith- 
metic, harmonic or geometric, so that in is always either the arithmetic, har- 
monic or geometric mean between a and b. As m is always the mean, its 
presence or absence in the calculations is indifferent ; and we are virtually 
dealing with two figures, a and b, and, treating them as equally important, 
are using even weighting. We may suppose that one of these terms, always 
a, when not equal to m, is given, and that this term is always larger than m 
(which is also a given term). Then the other, always smaller than m at the 
same period when a is larger, may always be expressed in these given terms. 
Thus when the terms are in arithmetic progression, 6 ^ 2m — a ; when in 

T_ . . , nut , . . . ^ m^ 
narmonic progression, o = ; when m geometric progression, o = . 

AiCt TYl CI 

We may suppose either (I) that the terms are equal at the first period and 
a and b become divergent at the second, or (II) that a and b are divergent at 
the first period and by converging become equal at the second. There will 
then be occasion for some explanatory and amplificative remarks. 

I. 

§ 1. In this supposition we have Oj = 7?i =: &i and a^ > m >■ b^. Here aj 
and &2 have varied /rom the mean to opposite extremes, a.^ having risen and 63 
fallen. 

The variation of o, from a, is always — ^ — . The variation of b^ from 

% m 

5, to the arithmetic extreme is 7^ ^^ ; to the harmonic extreme, 

b^ m 

62 2a — m a, , . &, a, m t.^. , . , . 

7- = ^ 7; ; to the geometric, 7^ = ;= — . JN ow the arithmetic 

Oi m Zttg — m b^ m Qo 

mean between the variations to the arithmetic terms is ^r ( — H "- )=^ 1 ; 

2 \ ?n m J 

wherefore these may be regarded as (simple) arithmetic variations (whose 

arithmetic mean indicates constancy). The harmonic mean between the 

524 



/ 

V 
\ 



ON COMPENSATORY VARIATIONS 525 

variations to the harmonic terms is :;— =; 1 ; wherefore these 

1 im iOj — m\ 

may be regarded as (simple) harmonic variations (whose harmonic mean indi- 
cates constancy). Tlie geometric mean between the variations to the geomet- 
ric terms is -^ -— = 1 I wherefore these may l)e regarded as (simple) geo- 
metric variations (whose geometric mean indicates constancy). 

§ 2. The percentage (in hundredths) of the rise of a., above m, reckoned in 

m, is always ~ . The percentage of the fall of b^ below m, likewise 

, , , . . , . , . . . . m — (2m — a~ ) a, — m 
always reckoned m in, in the aritlimetic variation is ^ ~ -= ~ ; 



ma^ 

m — „ '— 

., , . . . Za~ — m a, — m 

in the harmonic variation, ~ = ^ ; in the geometric, 

m zoj — m 

m 

= — . Thus, the percentage being reckoned in the mean or the uni- 
form starting point, it is only the rise and fall from the mean to the arithmetic 
extremes, or the arithmetic variations, that are in equal percentage. 

§ 3. But if we reckon the percentage in the terms reached, the percentage 

of the rise of a.,, in a^, is always — . The percentage of the fall of b^, 

, . , . , . , . ... m — (2m — a,) a, — m . , 

always in o„ in the arithmetic variation is ^ = ^ ; in the 

2m — a, zm — a. 



. . 2a» — m «, — m . , , . a, 

harmonic variation, = = — ; in the geometric, :;; — = 

■ma., aj m? 

2a2 — m a^ 

~ . Thus, the percentage being reckoned in the opposite extremes reached, it 

is only the rise and fall from the mean to the harmonic extremes, or the harmonic 
variations, that are in equal percentage. 

§ 4. If again we reckon the percentage always in the same direction, that 
is, (1) from the lowest terms, or (2 ) from the highest terms, or in other words, 
(1) from the starting point at the mean for the rising term and from the ex- 
treme point reached for the falling term, or (2) from the extreme point 
reached for the rising term and from the starting jioint at the mean for the 
falling term, we have the following percentages. (1) The percentage of the 

rise of a,, reckoned in m, as shown in ? 2, is ~ : and the percentages of 

m 

the falls of b^, reckoned in 6,, are given in § 3. We see that, the pc-centage 

being reckoned in the lowest terms, it is only the rise and fall from the mean to the 

geometinc extremes, or the geometric variations, that are in equal percentage. ( 2 ) The 

percentage of the rise of a^, reckoned in a,, as shown in § 3, is -^ ; and 

a^ 

the percentages of the falls of b^, reckoned in m, are given in § 2. We see 

that, the percentages being reckoned from the highest terms, it is only the rise and fall 



526 APPENDIX B 

from the mean to the geometric extremes, or the geometric variations, that are in equal 
percentage. 

II. 

§1. In this supposition we have cf^ > m > 6^ and a2 = w^ = ^2; and a^ 
and 62 have varied to the mean from opposite extremes, a^ now having fallen 
and 62 risen. 

The variation of a^ from a^ is always — = ~ . The variation of b.^ from b^, 

from the arithmetic extreme, is t^ = r : from the harmonic extreme, 

Oj 2m — Oj ' 

T- ^ = : from the geometric extreme, ~ = =: -^ . Thus 

Oi moi fli '^ 61 jji'' m 

2cti — m Oj 

these variations are the reciprocals of the preceding ; for we may suppose % 

here to be equal to ff2 there, and b^ here equal to b^ there. 

^ 2. Now unity is the harmonic mean between the variations from the 

arithmetic extremes ; for — = 1 ; wherefore the variations from 



1 / Oi 2m — OjN 
2\m m ) 



2' 

the arithmetic extremes to their mean are harmonic variations (whose harmonic mean 
indicates constancy). Unity is the arithmetic mean between the variations 

from the harmonic extremes ; for — ( 1 1=1; wherefore the vari- 

2 V Oi tti J 

ations from the harmonic extremes to their mean are arithmetic variations (whose 
arithmetic mean indicates constancy). But unity is the geometric mean be- 
tween the variations from the geometric extremes : for -\/— • — =: 1 ; where- 

* ttj m 

fore the variations from the geometric extremes to their mean are geometric variations 
(whose geometric mean indicates constancy). 

§ 3. In effect, if we reduce the terms at the first period, a^ and b^, to 
equality to m, and observe the same variations, we may reduce the terms at 
the second period, Og and 02, to figures no longer equal to m, as follows. We 

reduce % to m by multiplying a^ by ~ ; therefore we must reduce Oj in the 

aj 

, m m^ _,, , ... . , 

same manner ; but a, — = — . Ihus the variation 01 a^ to m is the same 

"ill/ "YYt 

variation as that of m to — . We reduce b-^ to m by multiplying &i by ^ ; 
and we must reduce b^ in the same manner. Now when the terms were in 
arithmetic progression, we had 6^ = 2m — o^ ; therefore we have 6^ • ^ = m • 

H = K • Thus the variation from 6,, when it is at the arithmetic 

2m — a 2m — % 

extreme, or from 2m — a, to m is the same variation as that of m to p, . 

2m — a^ 

Similarly when the terms were in harmonic progression, the variation from the 

harmonic extreme, b-., to the mean, or from '^ — to m, is the same as that 

la — m 

„ ??i(2a — m) .11, 

irom m to . And when the terms were in geometric progression, 



ON COMPENSATORY VARIATIONS 527 



the variation from the geometric extreme, b-^, to the mean, or from — to m, 

is the same variation as that of m to «i . Now — and ^ are the har- 

r(j 2)11 — «! 

monic terms around 7)t ; for - — ; ^ — -=m. Therefore varidtiovs 

1 / aj 2m — aA 

2 \n?'^ m^~ j 

from the (irithitietia extremes to their mean are the same as variatians from the mean to 
harmonic extremes. Consequently it is the percentages of these variaLions, 
reckoned in the extremes, the positions originally at the first period, but at 

the second i)eriod in the reductions, that are equal. And — and 

«i «i 

are arithmetic terms around m ; for half the sum of these is in. Therefore 
rariations from the harmonic extremes to their mean are the sume as variations from 
the mean to arithmetic extreines. Consequently it is the percentage of these vari- 
tions reckoned in the mean, the position originally at the second period, but 

at the first period in the reductions, that are equal. But — and Oj are geo- 

jm'^ 
metric terms around m as their mean ; for \ — ■ o, =: w; . Therefore varia- 

' «j 

tionsfrom the geometi-ic extremes to their mean are the same as variations from the 
inean to geometric extremes. Consequently it is the percentages of variations 
reckoned either from the top or from the bottom that are equal. 

§ 4. Thus if two quantities, equal at first, vary to the opposite arithmetic 
extremes, and then vary back to the same figure as at the start, their first vari- 
ations, diverging, are arithmetic variations, and their later variations, con- 
verging, are harmonic variations. If two quantities, equal at first, vary to 
the opposite harmonic extremes, and then vary back to the same figure as at 
the start, their first variations, diverging, are harmonic variations, and their 
later variations, converging, are arithmetic variations. But if two quanti- 
ties, equal at first, vary to the opposite geometric extremes, and then vary 
back to the same figure as at the start, both these diverging and converging 
variations are geometric variations. 



III. 

The following explanations may be offered. 

§ 1. In I. we have been considering variations of states which may be 
represented thus, 

ag > m > 62 



b. 



= 1. 



But we have been averaging the variations, with even weighting. Hence, in 
accordance with Appendix A, I. § 7 (II. §6, III. §6, and V. §2), we have 
been performing the same operation as if we averaged the figures at each 
period separately, with even weighting in each case, and then compared the 
results. 

§ 2. In II. we have been considering variations of states which may be 
represented thus, 



528 APPENDIX B 

Cg =^ m = 52 "^ 
% > m > 6j ??i 



in eacli of which averages even weighting is employed. But we have been 
averaging the variations with even weighting, and now the operations are not 
the same in two out of the three cases. 

§ 3. Thus if the terms change from the arithmetic extremes, applying the 
arithmetic average with even weighting to the figures at each of the periods 
separately, we have 

A2 J(m + m) '"" 



Ai i^ai+ (2m — ai)| 



1. 



But we do not obtain this result by arithmetically averaging the variations 
with even weighting ; for 



1 / ??i in \ w? 

2 V 0^ 2m — ctj / a^(2??i — a-j) ' 



but only by harmonically averaging them, if we are to use even weighting, 
as shown in II. § 3. This is because the comparison of the arithmetic aver- 
ages with even weighting, when the figures are unequal at the first period, is 
the same as the arithmetic average of the variations, not with even weighting, 
but with weighting according to the sizes of the terms at the first period, 
according to Appendix A, II. | 7 ; 1 and, according to Appendix A, IV. § 2, 
this is the same as the harmonic average of the variations with even weight- 
ing. 

§ 4. If the terms change from the harmonic extremes, applying the har- 
monic average, we have 

1 



2 \m m/ 



= - = 1 



A2_ 

Ai 1 J 

2 \ ^2 "^ ma^ ) 

And again we do not obtain this result by harmonically averaging the varia- 
tions ; for 

1 m(2ai — m) 



1 ('hA__J'T^^\ 

2 V™ 2ai — m/ 



«i^ 



but only by arithmetically averaging them, if we are to use even weighting, 
as shown in II. § 3. This is because the comparison of the harmonic aver- 
ages with even weighting, when the figures are unequal at the first period, is 
the same as the harmonic average of the variations, not with even weighting, 
but with weighting inversely according to the sizes of the terms at the first 

^ Here, with even weighting, with a limited between m and 2m, the result is 
larger than unity, indicating a rise. This is because the avei-age of the variations 
can be brought down to unity only by giving greater weight to the fall of a-^. 



ON COMPENSATORY VARIATIONS 529 

period, according to Appendix A, III. § 7 ; '■' and, according to Appendix A, 
IV. 'i 5, this is the same as tlie arithmetic average of the variations Avith even 
weighting. 

2 5. But if tlie terms change to tlie geometric extremes, we have 

Aa V »i ■ m _ m 

Ai ~ / ^ ~~m~ ' 
^ a, 

and we obtain the same result by geometrically averaging the variations with 
even weighting, as shown in II. ^ 3. This is because the comparison of the 
geometric averages is the same as the geometric average of the variations with 
tlie same weighting, according to Appendix A, V. § 5. 

IV. 

The consequences indicated in II. {< 4 may be amplified as follows. 
§ 1. Suppose we have these three series at three different periods : 

1st period — o^ = m = 6]^ ; 
2d " a2>m>62; 

3d " Os ^ ■"* = ^3 ; 

in which a-^ = a^ and b-^ = b^. Here we have two variations of the series, 
which, with substitution of a^ for Oj and of ftj for 63, are 



1st variation- 
2d 



aj > m ]> 62 
a^ = m = 61 ' 

o^ = m = 61 

€12 ^^ ™ ^ ^2 ' 



But the two variations together i)roduce a return to constancy ; for 



-b. 



= =1, 



For what follows, however, it is not necessary that o., and b.^ be on opposite 
sides of VI. They may be any quantities, with any variation of their mean 
( now to be represented by m.^) from m^. All that is needed is that they return 
at the third period to what they were at the first ; whereupon also their mean, . 
//*;!, will return to wij. 

I 2. If we average these series separately and then compare the averages, 
we get inverted results, Avliich are reciprocals of each other. Then the com- 
parison of the third period witli the first by means of the two intermediate 
comparisons— which comparison consists in multiplying together the results 
of those comparisons — turns out correctly ; for these reciprocal results multi- 
plied by each other give unity. 

1 3. We have two sets of variations, which are 

2 Here, with even weighting, with a larger than on, the result is smaller than 
unity, indicating a fall. This is because the average of the variations can be 
brought up to unity only by giving a greater weight to the rise of 6j. 

34 



530 



APPENDIX 


B 


1st set of variations — 


«2 h. 

' — ' zT ' 


2d " 


a, 0, 



and tlie whole variations from the first to the third period are — =: ^ = 1 and 

a-i «! 

=^ = r^ ^ 1 : wherefore the average of these whole variations is always unity, 

indicating constancy. But if we average each set of variations, and seek thence 
the average of the whole variations, we obtain the following results. 

I 4. The arithmetic averages of the variations, with even weighting, are, 

of the first set, 77 ( — + y^ ) , and of the second, 77 ( — + 7^ ) . The multipli- 
' 2 Voi 61/ ' 2 Voj 62/ 

cation of these gives 77 + -r ( -M H — V I > which by elimination of a, and 6„ 
^ 2 ' 4 Va2*i 01*2/ 

which are equal, reduces t0 7j + -r( — + r^)' This does not reduce to unity 

unlesss a^^h^. In all other cases its result is above unity,'' indicating a rise, 
instead of constancy. 

§ 5. The harmonic averages of the variations, with even weighting, are, of 

the first set, - — ; r*^ , and of the second - — ; j-^- . The product 

2V«2 ^2/ .2\a^^bJ 

of these is ^ z. — 7-^ , which does not reduce to unity except, as be- 

2 "^ 4 V *2 «2 / 

fore, in the simple case when aj = h^. In all other cases its result is below 
unity, indicating a fall, instead of constancy. 

§ 6. The geometric averages of the variations, with even weighting, are, of 

the first, A/ - • r^) and of the second, \ • -r- The product of these is 

unity, indicating constancy. Therefore the continuous use of the geometric mean 
through both the variations gives the right result for the two variations together. 

§ 7. But again, if we used the arithmetic average of the variations for the 
first set and the harmonic for the second, or the harmonic for the first and the 
arithmetic for the second, all with even weighting, the product of the results 
thereby obtained give unity, indicating constancy. Therefore the alternate 
use of the arithmetic and of the harmonic means of the variations gives the right result 
for the two variations together. 

This proposition is plain if we remember that the harmonic average of 
given figures is the reciprocal of the arithmetic average of their reciprocals, 
and reversely ; and that when variations take place from the mean to the 

^ To demonstrate this it is necessary to show that 7- + - > 2. Now t + - = 

a b a 

z — . Let this equal x. Then ; 7- = x — 2; whence j — = a; — 2. 

ab ab ab ab 

(a— 6)2 
But ^^ r~— is always positive, a and b being positive. Therefore a; — 2 is posi- 
tive, and X is greater than 2. 



ON COMPENSATORY VARIATIONS 531 

arithmetic or to the harmonic extremes, the return variations from these ex- 
tremes to the mean are tlie reciprocals of the first variations. 

V. 

§ 1. Lastly we may analyze the variations in the following series : 

1st period — aj > m > 7>j ; 

2d " «2' = ^)<»«<*2(="i) ; 

in which m represents any one of the three means. Here the figures are sup- 
posed simply to change places, the one falling to the other and the other ris- 
ing to it. The variations are ^ and ,\ which are reciprocals of each other. 

§ 2. It is plain that when «, is becoming equal to b^ , it must pass through 
the mean between them, that is, it must fall first to the mean and then from 
the mean ; and when Jj is becoming equal to a^ , it must pass through the 
mean between them, that is, it must rise first to the mean and then from the 
mean. 

§ 3. Therefore if the mean be the arithmetic, the approaches of a^ and b^ 
to this mean are harmonic variations (above, II. § 2), and their departures 
from it are arithmetic variations. 

If the mean te the harmonic, the approaches of a^ and b^ to this mean are 
arithmetic variations, and their departure from it are harmonic variations. 

But if tlie mean be the geometric, both the approaches to it and the de- 
partures from it are geometric variations. 

^ 4. Therefore also the whole variations of o-i to b^ , and of b^ to a^ , are 

geometric variations. This is plain also from the fact that the variations are 

reciprocals of each other. It may be shown, further, in the following way. 

The percentage of the rise of b^ to Oj may be represented as a rise by p per 

cent, (reckoned in b■^). Then the percentage of the fall of a^ to b^ (reckoned 

p 
in rtj), being a fall by p from 1 -f- p, is a fall bv ^ 7— per cent. The rise of 

I -t p 

61 is the same variation as the rise of 1 to 1 -f- p , and the fall of f(j is the same 

pi . . 

variation as the fall of 1 to 1 — , = , ; that is, the rise and the fall 

l-fp 1 -hp' 

are from 1 to geometric extremes, and are geometric variations. 

§ 5. Moreover all geometric variations in opposite directions are expres- 
sible as rises and falls to reciprocals, that is, as rises from 1 to 1 -|- j3 and as falls 

from 1 to . Therefore all geometric variations in opposite directions are 

the same as variations from some one figure to another and from the latter to the for- 
mer. 

§ 6. Still, in these geometric variations, a-^ in becoming b^ and b-^ in be- 
coming Oj must pass through the arithmetic and the harmonic means. There- 
fore all geometric variations may be viewed as composed of two variations, the one 
harmonic and the other arithmetic, in either order as we please. 

§ 7. If a and b vary at the same uniform speed so that «i becomes 6j and 
fcj «! in the same time, they pass through the arithmetic mean at the same 
moment, which is the halfway point in the time for the whole alternation. 



r)32 APPENDIX B 

If a and b vary at the corresponding uniform rates so that a^ becomes b^ and 
61 a^ in the same time, they pass through the geometric average at the same 
moment, which also is the timal halfway point. There is a uniform varia- 
tion which will carry them through the harmonic mean at the same moment 
at the timal halfway point. 

§ 8. If the variations be the ones at the uniform rates, with a and b pass- 
ing through the geometric mean at the timal halfway point, the rising figure, 
b, crosses the harmonic mean first and the arithmetic mean last, and the fall- 
ing figure, a, crosses the arithmetic mean first and the harmonic mean last. 
At every moment a and b are at geometric extremes around their halfway 
point. Therefore (because of Appendix C, VI. ? 2) at the moment when b 
crosses the harmonic average, a crosses the arithmetic, and at the moment 
when b crosses the arithmetic, a crosses the harmonic. 



APPENDIX C. 

REVIEW AND ANALYSIS OF THE METHODS EMPLOYING 
ARITHMETIC AVERAGING FOR MEASURING VARI- 
ATIONS IN THE EXCHANGE-VALUE 
OF MONEY. 

Arithmetic averaging is the common feature in so many widely differing 
methods of calculating variations in the exchange-value of money from the 
variations of jirices, that it may be well to review these methods l)y expound- 
ing them all in the same system of notation ; for many liave been made with- 
out any formulation at all, and the others have been formulated with so many 
difl'erent symbols that comparison is difficult. Also some of the methods are 
so complex that analysis is necessary to disclose their real nature. 

The notation here employed is that employed in the body of this work. Prices 
of the classes [A], [B], etc., are represented at the first and second periods re- 
spectively by numbered Greek letters — Oj, Wj, /^j, /^2) • These, unless other- 
wise specified, are prices of the ordinary mass-units used in commerce. When 
prices are taken at the first period all as 1.00 and the prices at the second 
period are reduced accordingly, the Greek letters are accented, thus, «/ =^ 

1^/^ (=;1.00), and a/, ji./, (these figures by their excess above, 

or deficiency below, 1.00 directly indicating the variation of the prices ; for 

a/^ -^, p/ = ^', and so on). The numbers of these mass-units (or masses 

whose prices are used) in the classes respectively, as they are supposed to oc- 
cur in trade, or amounts all reduced in the same proportion, are represented 

by j:, y, , ov x^, y^, , (which are the same symbols as in Appendix 

A, while a, j3, here take the place of a, b, as there used). These 

letters, x, y, , we must notice, do not represent the weights of the classes 

(except in the averagings, generally eliminated, at the difl'erent periods sep- 
arately ; they never represent the weights of the price variations). The 
weights of the price variations are represented by a, b, for the varia- 
tions of the prices of the classes [A], [B], respectively. These letters, 

a, b, , represent, supposedly, the numbers of ideal individuals in the 

classes, or amounts reduced in the same proportion, — that is, the relative sizes 
of the classes measured by their exchange-value (or money-value) impor- 
tance. In other words a = xa, b = yl^, and so on, or a : b : : ." xa '.yfi 

533 



534 APPENDIX C 

without specifying at which period, or how, the prices are used. The num- 
ber of classes used in the calculation is represented by n. The number of 
actual individual prices (those of the mass-units employed) in all the classes 

together, by n^ ; so that n^ ^x-\- y-\- to n terms. The number of ideal 

individual prices in all the classes together, by n''^; so that ti^'' = a + b -|- 

to n terms. The terms x, y, , a, b, , n, n' and n", when the 

same for both the periods compared, need not be numbered. But when they 
are employed distinctively for the conditions at one period alone, they are 
also to be distinguished by subscript numbers. 

I. DUTOT'S METHOD. 

Dutot, 1738 ; see B. 2, especially pp. 370-373. 

The prices of articles are taken at their market quotations on the mass- 
units that happen to be employed by merchants, however large or small, and 
are added together, the sums for the two periods being compared. It may be 
expressed thus, 

P2 S+/52 + 



Pi ai + ^i + 



Here only sums are used. But we know that this gives the same result as 
if the arithmetic average were drawn for each period (see Appendix A, II. 
\V). We also know that it contains hidden weighting for the price variations, 

viz. according to «i, /J^, , with arithmetic averaging, (see Appendix A, 

II. §7). Therefore this method contains what we have called haphazard 
weighting (in Chapt. IV. Sect. II. ^2). 

This method has since been employed by Levasseur, B. 18, pp. 179, 180, 
and by A. Walker, B. 27, even in series of many years ; by a writer in the 
London Bankers^ Magazine, B. 37 ; by Krai, B. 98 ; and by Fraser and Sergei, 
B. 120. 

( As the mass-quantities — the mass-units hit upon — are constant, this method 
is really a variety of Scrope' s method, to be treated of later. But as it was 
invented and has been used without any reference to the mass-quantities, it 
deserves to be classed by itself. ) 

II. CARLI'S METHOD AND ITS VARIETIES. 

§ 1. Carli, 1766 ; see B. 4, especially pp. 350-351. 

All the prices being taken at the first period as equal to 1.00, the prices at 
the second period are taken on the same masses, and the percentages of their 
variations noted ; then the average is drawn between these. The procedure 
in full is described in the following formula, upon the prices of any mass- 
units. The average percentage is represented by %. 



0/ ^ 
/o- 



100 f a, — a, (3, — (3, \ 

n \ a, ^ [3, + ;' 



in which the answer indicates a rise if positive, a fall if negative, and con- 
stancy if zero. 

The same result is yielded by the following formula, 






REVIEW OF METHODS 535 

in which the answer indicates constancy if unity, a rise if above unity, a fall 
if below unity ; and indicates the percentage of the rise in the second decimal, 
and the percentage of the fall in the second decimal of the remainder when 
the answer is snbftracted from unity. If the prices at the first period are all 
taken at 1.00, and those at the second period are reduced accordingly, the 
same result is given, in the same way, by the following, 

f = li< + P/ + )' 

which is practically the form employed by Carli. Or if the prices at the first 
period are all taken at 100, by this, 

^=^(100«/ + 100/3/+ ), 

in which the answer indicates constancy if 100, and the percentage of a rise 
or fall according to the units it is above or below 100. 

This last form of the method — the one which has usually been employed 
— was introduced by Evelyn, 1798 ; see B. 5. This variation of Carli' s 
method is only formal, and, when it is employed, the method still deserves to 
be called Carli' s method. In it, it is plain that when we are comparing 
only a second with a first period, there is even weighting of the price varia- 
tions. 

§ 2. Evelyn, however, compared several periods all with the same first 
period (some of the second periods being later and some earlier than the com- 
mon first or basic period) ; and this also has become the usual practice. Now 
in this case — confining our attention to the simplest form of the formulae, that 
in which prices are reduced to 1.00 at the basic period, — after comparing a 
second period with the first and getting this result, 

|=l(a/ + /3/+ ), 

and comparing a third period with the same first, with this result, 

f:=^<+'v+ ), 

if we compare these with each other, our comparison of these is now as fol- 
lows, 

P. ^ < + /?./+ 

P2 <+P/+ ' 

or 

F,^a ,^l3,^ 

P2 «2,^2, ■ 

«1 /?! 



But here we know that the weighting of the price variations is no longer even, 

— it is accidental, being according to a/, (3/ , or -, ^, , that is 

according to the price variations between the base and the earlier period. 



536 APPENDIX C 

Thus this method — mostly without the knowledge of those who employ it — 
switches off into something quite different. 

This method of Carli was so employed by Porter, B. 11 ; and, with Eve- 
lyn's formal variation, it has been employed by J. P. Smith, B. 7 ; the Econo- 
mist, on the lists of prices begun by Newmarch, B. 19 and 20 ; Laspeyres, B. 
25 and 26 ; Bourne, at first, B. 46 ; Burchard, B. 53 ; Hansard, B. 67 ; 
Sauerbeck, B. 79-90 ; Rogers, B. 92, pp. 787, 789, 790, 791 ; Falkner, partly, 
B. 111-113 ; Wiebe, partly, B. 124, pp. 369-382 ; Whitelaw, B. 130, pp. 32- 
33 ; Wetmore in B. 119 ; Barker, B. 128 ; Powers, partly, B. 131, pp. 27, 28, 
29, etc. Geyer employed it with another formal variation, reducing the first 
prices to about 2.00, B. 23, pp. 321-322. 

§ 3. Evelyn introduced furthermore a real variation. He comited always 
wheat by itself, butcher' s meat by itself, and day labor by itself, all as equally 
important, and also as equally important with each of these a figure made up 
of the prices of twelve other agricultural products, each of which had been 
counted as of equal importance in making up this figure. This is irregular 
weighting by classification (in which classes count more the fewer there are in 
the divisions). It is not worth while to give a formula for this. It would be 
exceedingly complicated if we attempted to find the weighting employed in 
the comparison of two later periods with each others. Whenever, with the 
rest of Carli' s method, instead of absolutely even weighting in the comparison 
of the other periods with the basic one, there is employed such irregular 
weighting by classification, this may be called Evelyn' s method, as it employs a 
feature really distinctive in Evelyn's procedure. 

This method was sometimes emjaloyed by Young (who in doing so aban- 
doned his own method), B. 6, pp. 84, 118 ; also by Soetbeer, B. 19 ; by Krai, 
B. 98 ; and by Wiebe, partly, B. 124, pp. 383-386. 

III. YOUNG'S METHOD AND ITS VAEIETIES. 

? 1. Arthur Young, 1812 ; see B. 6, p. 72. 

Prices at the first or basic period are taken at 100 and at the other periods 
reduced accordingly. Percentages of the variations are used, and these are 
multiplied by the weights (according to the relative total exchange-values of 
the classes, in general, at no particular period ) , and the sum of the products 
divided by the sum of the weights. Thus 

100 / a, — a, , , /3„ — /?, , \ 

^^=^(^■^^ + ^^^3^"+ to. terms), 

in which the answer has the same meaning as in Carli' s. 

The same result, with our usual interpretation of the answer, is yielded 
by the following, 



' = ;7^H;+^t + ton terms); 



or, if we start with prices at 1.00, by 

P 1 

p^ = — , (aV + "^(^2^ -\- to n terms). 

Either of these may be regarded as the formula for Young's method. 



REVIEW OF METHODS 537 

This method was approved and recommended by Lowe, B. 8 ; and has since 
been reinvented, with slight variations in tlie procedure, by Ellis, B. 36, and 
Wasserab, B. 105 ; by Sauerbeck at times (using the weights of 1889-1891), 
B. 82, p. 218, and later; by Falkner at times (using the weights of periods 
at the end of the series), B. 112, p. 63 ; by Fonda, B. 127, pp. 160-161 ; and 
a correct formula has been made for it by Westergaard, B. 110, p. 220, in this 
cumbrous form, 

Po ^ a^ a, b^ (3^ 

Pi~a + b + ■ ai a + b-H 'A ' 

which easily reduces to the simpler form above given. 

§ 2. Young himself happened not to do so, but the rest of these writers 
have employed, or recommended, this method in a series of periods, all the 
later being compared directly with the first. Then a comparison between any 
of the later periods would be as follows, 

P, _ sia / + \i(3/+ 

P, aa/ + b/3/4- ' 

or, with the prices of any mass-units, 

^i ^ «i Pi 

in which, as we know, the weights of the price variations are no longer a, b, 
, but a«2^! b/V. ) ora- , b A-% (see Appendix A, II. § 8). 

For this difference between the comparisons of the later periods with one an- 
other and the comparisons of each of the later periods with the first, Wester- 
gaard has found fault with this method (see here Ch. V. Sect. VI. ?, 7). 

§ 3. Variations upon this method have been made by GifFen and by Pal- 
grave. Giffen's variation is mostly formal, though it is real as regards the 
choice of weighting. The formula for it, for the second period, is 

% =— ^ \3^.^{a/ — 1) -f- h^{|3./ — I) -\- to n terms}, 

or, with ordinary prices, 

(A formula for it somewhat like the last, but not specifying the period at 
which the weights are chosen, is given by Edgeworth, B. 59, p. 265. ) It is 
confined to custom-house returns, and so, like James's method already (see 
next section ), has the fault of representing the relative weights only of goods 
exported or imported. He makes one calculation each for exports and for 
imports. His weights, which are according to the total money-values of the 
classes, are reduced to percentages of the total money-value of all the exports, 
or imports, not at the basic period, 1861, nor on an average, but at another 
period, 1875, the seventh in his series (which skips every other year). These 



538 APPENDIX C 

percentages should add up to 100 ; but as he could not manage all the classes, 
his weights for exports in that year added up only to 73. 1 and those for im- 
ports only to 84.38 ; wherefore these figures are taken instead of 100 for his 
starting points (in 1861, the prices then being reduced to unity). [He said 
it is unimportant of what year he chose the weighting, because the totals of 
the percentages varied very slightly — between 72.7 and 80.4 for exports, and 
between 82.42 and 88.88 for imports. B. 40, p. 8, cf. B. 39, p. 5. This is 
no reason at all, as it overlooks that the particular weights of the individual 
classes varied considerably.] In extending his investigations backwards, in 
B. 36, he had to confine himself to fewer classes still, so that the figures (for 
the weights of 1875, and the price averages of 1861) reduce to 65.8 for ex- 
ports, and to 81.16 for imports. (He describes his method best in B. 45, qq. 
709-716. ) 

§ 4. Palgrave, B. 77, has introduced the real variation of employing the 
weighting of every subsequent year in the comparison of it with the first or 
basic period ; and he appears to have been followed by Sauerbeck in one of 
the latter' s "tests" (see B. 82, p. 218). Prices are taken for the basic period 
at 100 ; but here, for simplicity, we may take them at 1.00. The formula for 
comparing the second period with the first is 

P 1 

^ = —„ (aa^z^ + ^ii^-i + to »* terms) ; 

or, adapted to the prices of any mass-units, 

P; 
P 

which in full is 



\'^h\^''X'^^'\'^ tonterms). 



P; = ^^("^"^a"; + ^^^4;+ to^terms), 

(like this it is given by Edgeworth, B. 59, p. 265), which reduces to 

P;^<r< + ^^# + toutermsj. 

When dealing with a third, or any subsequent, period, Palgrave compares 
it with the first, in the same manner, thus, 

P 1 

-i = —,A ^f-i + bs/^s^ + to n terms) . 

j^i % 

Therefore a comparison between the results so obtained, as representing a 
comparison between the subsequent periods themselves, means a comparison 
in this form, 

-p • — , (a^as^ + bg/Jg^ -f -to n terms) 

fj ''h 

P 1 

"" —„ (ajOa^ + b2^2^ + to n terms) 

which, by restoring the values of n^" and n.^', and converting, reduces to 

Ps ^ asgg" + bg/^s' + a2 + b2+ 

P2 a2< + b2/?2' + ' a3 + b3+ ' 



REVIEW OF METHODS 539 



and the same may be more generally stated thuss, 

^2 o "=2 I >, /^2_L 'a, + b3 + 

or, lastly, in this form. 



^2 X^4-V^-l. ^3«3 + 2/3/^3 + • 

The inverse of this formula gives 

a 2 /J 2 

2 I r2 I 

itf 03 _ x^a, + 2/3/?, + ^2 «! "•" ^2 ^^' -I- 



ilf02 ^202 + 2/2^2 + ,«3V„^V 

which we recognize as a formula with double weighting (with very curious 
weighting in the separate averages at each period). 

IV. SCROPE'S METHOD AND ITS VARIETIES. 

? 1. Scrope, 1833 ; see B. 9. 

The method there described may be thus formulated, 

p ^(•c«2 + y/^2+ ton terms) 

■t 2 n 

P ~~ 1 ' 

' ^ ( xflj + 2//5j + ton terms) 

this representing the variation in the " mean " ; or " in the sum " as follows, 

F^^ xa^ + ylS^-'r 

Pi a-ai + y/?i+ " 

Scrope did not specify at which period, or how, the mass-quantities are to 
be chosen. All he had in mind was a general average of some sort. 

We know that this method is the same as the arithmetic average of the 

price variations with weighting according to xa.^, y/S^, (Appendix A, II, 

I 8), and as the harmonic average of the price variations with weighting ac- 
cording to xa^, y[3^, (Appendix A, IV. § 3), and, in some cases, as the 

geometric average of the price variations with weigliting according to the 
geometric means between these weights (Appendix A, VI. ^^7-8). As it 
virtually employs arithmetic averages of the prices at each period, it is simpler 
to regard it as the arithmetic average of the price variations with the weight- 
ing of the first period, and it could be expressed thus, in the comparison of 
the second period with the first. 



P, 
P, 



i-; = V'(^i«-; + ^4;+ tonterms). 

In this form it might be regarded as a variety of Young's method, and it was 



540 APPENDIX C 

originally no doubt suggested without an idea of its being other than Young's 
method. Bat it is essentially different. 

Its differentia from Young's method is that it uses mass-quantities, with 
hidden weighting of the price variations, instead of direct weighting accord- 
ing to total money-values. This rests on the difference that it compares 
price averages (or price sums) at two or more periods, instead of directly 
averaging price variations between two periods. 

But just as Young's method uses single weighting in every average of price 
variations, so Scrope's method uses the same mass-quantities at both periods 
in every comparison. 

Used, as recommended by Scrope himself, in a series of years, always on 
the same mass-quantities, if the total sum of the money-values at the first 
period is reduced, as usually is the case, to 100, then the others are to be 
reduced accordingly merely by dividing them by the same figure. Suppose 
this figure is r. Then the index-numbers i 1 ) for a series of years are 

Ii = 7(^«:+2//5x+--) = 100, 

\^~{^a,^yP,+ ), 

and so on. A comparison between any of these index-numbers obviously 
yields the same result as a direct comparison between the corresponding 
periods (using the same mass-quantities). 

This method, with more or less undefined mass-quantities, has been recom- 
mended and formulated by Fauveau, B. 54, p. 356, and by Walras ( who 
calls it the method of "the multiple standard"), B. 69, jjp. 15-16, B. 70, 
pp. 432, 468, B. 71, p. 131 ; and has been recommended by Newcomb, B. 76, 
p. 211, Andrews, B. 107 and 108, J. A. Smith, B. 129, pp. 27-29, Pomeroy, 
B. 135, p. 332, and Parsons, B. 136, pp. 134-136. 

^ 2. Definiteness in regard to the mass-quantities may be given to this 
method in several ways, among which three have been noticed ; ( 1 ) by em- 
ploying the mass-quantities of the first (or basic) period ; ( 2) by employing 
the mass-quantities of the other periods which are compared with the first (or 
basic) period ; or (3) by employing some mean between these methods, two 
such having been suggested. All these have been formulated, most of them 
recommended, and some of them applied. 

The first two were first distinguished and formulated by Drobisch, B. 29, 
pp. 36-39, who added notice also of one form of the third, B. 31, pp. 423- 
425, but who rejected them all, except in an accidental case when it happens 
that the mass-quantities are the same at both the periods compared. They 
have all been similarly condemned by Wicksell, unless it accidentally hap- 
pens that this method tried in the first way and in the second way yields the 
same result, B. 139, pp. 11-12. On the other hand all three were consid- 
ered equally good by Sidgwick, who therefore regarded none as authoritative, 
B. 56, pp. 67-68, (cf. Edgeworth, B. 59, p. 264). Similarly Padan, B. 141. 



REVIEW OF METHODS 541 

But Lindsa>', likewise considering all three equally good, and wanting only 
one to be employed, is indifferent which be adopted, B. 114, pp. 25-26. 

(1) The method with the mass-quantities of the first or basic period was 
formulated and recommended by Laspeyres, B. 26, p. 306 (who seems to have 
been unaware that Drobisch had already formulated and condemned it). 
Adapted, and modified (by omitting use of 100), his formula is 



Pj _x^a^ + y^li^ + 



In a series of years later comparisons would be in this form, 
P3 _ Xjag + 2/,/?3 + 



P2 ^Y'2 + Viih + 



in which the weighting of the price variations (arithmetically averaged) is 
rather curious, being according to the money-values of the mass-quantities of 
the first or basic period at the prices of the earlier of the two periods com- 
pared. 

This variety of the method has been used by Bourne, in one of his opera- 
ations upon custom-house returns, B. 47-49. The next variety, as used by 
Sauerbeck for periods prior to his basic period, is like this in using the mass- 
quantities of the earlier periods (but not those of the basic period), (cf. Edge- 
worth, B. 59, p. 264). 

(2) The method with the mass-quantities of the later periods (or, more 
generally, of the periods other than the basic) was first consciously employed 
by Paasche, B. 33, pp. 171-173, who has been followed by v. d. Borght, B. 
55, and Conrad, B. 96 and 97. The same method has also been employed, 
in a very general way, and dogmatically, by Mulhall, B. 74, p. 1157, B. 75, 
p. 1 ; and in his second "test" by Sauerbeck, B. 82-90, who is, partly, fol- 
lowed by Powers, B. 131, see p. 29. 

The formula for the method in this form is 

P2 _ 2:202 + 2/2/^2 + 



Pi ^2«l + 2/2/^1 -I- 



(cf. Edgeworth, B. 59, p. 264). Thus the weighting in (arithmetically) aver- 
aging the price variations is according to the money-values of the mass-quan- 
tities of the other period at the prices of the first or basic period. 

In a series of years, a comparison between any two of the later years would 
be as follows, 



P3 _ ^3«1 -^ J/S-^I + 
P2 a:2«2 H- 2/2/^2 + 



^3«3 + 2/3/^3 4- a-gfli 4- ya/J) 4- 



a;2«2 + 2/2/^2 4- ^sP-V+Vi^l 

which uses double weighting in a curious form (the weighting of the average 
at each period compared being according to the money-values of its mass- 



542 APPENDIX C 

quantities at the prices of the basic period). If the prices have all been re- 
duced so that they are at unity (or 100) at the first period, the mass quantities 
being correspondingly (inversely) altered, the formula becomes 

T,^ x/a/ + y/p/+ x/ + y./+ 

P2 ^■zW-\-y/l3/ + %' + 2//+ • 

This method had already been employed in a particular application to 
custom-house returns, in England by James, B. 10, (his method being based 
upon the peculiarity that British exports were officially rated always at the 
same prices at which they had been originally rated in 1694, ' and that since 
1798 there were added to the records also the "declared values" according to 
the current prices set upon their goods by the merchants themselves, so that 
his weighting for each year was according to the money-values of its mass- 
quantities at the prices of 1694) ; and in France (on the " official values " 
based on the prices of 1826 and the "actual values " recorded after 1847), both 
for imports and exports, separately and combined, in one jump from 1826 to 
1847 and thereafter annually to 1856, by Levasseur, B. 18, pp. 181-184, 188- 
192. This work was resumed by De Foville, B. 50, who continued it down 
to 1862, when the "official values" on the old prices of 1826 were discon- 
tinued. (Thereafter the French Custom-house every year first published 
"provisional values " consisting of this year's mass-quantities at the last 
year's prices, and later the "actual values" consisting of this year's mass- 
quantities at this year' s prices. To compare the latter with the former for 
every year is always to do the same operation as here indicated in the formula 

p 
for ^ , a new basis being employed every time. De Foville continued his 

series in this way down to 1880, stringing them out in a series following upon 
the previous series. ) 

( 3 ) A mean between the first two varieties may be by merely using each 
separately and drawing the (evenly weighted arithmetic) mean between their 
results. The operation is expressed by this formula, 

P2 ^ 1 / a^]Q2 + yA+ _|_ ^2«2 + y2/^2 + \ 

Pi 2 \x,a,+y^fi,-\. -^^^a, + y,p,+ J' 

This appears to be the form Drobisch had in mind. ( It is so understood by 
Edgeworth, B. 59, p. 265. ) 

Another is to form one calculation on the (evenly weighted arithmetic) 
mean of the two mass-quantities of every class, according to this formula, 

^ ^ i(a;i+a;a)«, + |(yi+y.J /33 + 

Pi i-[Xi+X2)ai-\-^y^-\-y^)(3^-{- ' 

This formula, which reduces to 

P2^(^i + a:2)«2+ (yi + y2)/^2 -t- 

Pi (a;i-f 0:2)01 4- (2/1 4-2/2 '^1 -t- ' 

^ Or 1696 or 1697 ; of. Flux, B. 140, p. 67, who has recently employed the same 
method. 



REVIEW OF METHODS 543 

was independently suggested by Marshall and Edgeworth (according to the 
latter, B. 51, p. 265), and is recommended by the latter, ibid., p. 266 (but 
not decisively, cf. p. 255), and by the British Association Committee, First 
Report, B. 75, pp. 249-250. Lindsay recommends it when comparing periods 
long separated, B. 114, p. 25. It is formulated by Zuckerkandl, B. 115, p. 
248, B. 116, p. 247. And it is experimented with by Powers, B. 131, pp. 
28-29. 

As regards the relationship between these two sub-varieties, it is obvious 
that if in the formula for the first the denominators happen to be equal as 
wholes, this formula reduces to the formula for the second ; but not otherwise. 

The arithmetic average of the mass-quantities over the whole epoch in- 
vestigated (in his case of thirty five years) has been recommended as the best, 
and employed, by Powers, B. 131, p. 24. 

Another mean of the two mass-quantities of every class is the evenly 
weighted geometric. Scrope's method applied to such mean mass-quantities 
is recommended in this work (Cliapt. XII.) under the name of Scrope's 
emended method. Its formula is 

P^ _ ayx-^x^-^yyy^y^ -t- 

^ 3. It may here be added that it would be obviously absurd to attempt to 
use the mass-quantities at both periods in the following way, 

P.; _ x.fi^ -f y^/J^ -I- 

Pi ^i«i + 2/A + ' 

for this would be merely a comparison between the total valuations or inven- 
tories at the two periods, which might be altered by a change in the mass- 
quantities without any price variations. But if we suppose that the mass- 
quantities have not changed, i. e., that x^ = Xi, y.^ = y^, and so on, this formula 
would be the same as any of the preceding. On the supposition that the 
mass-quantities have remained nearly the same, it is employed by Edgeworth, 
B. 59, p. 272, cf. pp. 264, 293 ; also, inverted for the exchange-value of 
money, by Nicholson, see here below, V. ^ 3. Furthermore, if the mass- 
quantities all change alike, so that ^ ^ — = = r, this formula would 

^'i 2/i 
also be serviceable if we multiply the denominator by r, in order to counter- 
act the change of the mass-quantities ; for then 

Pj^ g:2"2 + 2/21^2+ 

Pi r(a;,ai + 2/i/3i -t- ) 

reduces to 

P2 ^ a^2«2 + V i ft-l + 

Pi a;2ai+2/.A+ ' 

or if we multiply the numerator b^' - ; for then it reduces to 

P2 _ a^l«2 + 2/2/^2 + 



Pi ^'i^i + 2/1/^1 4- 



544 APPENDIX C 

and these two reduced expressions are equal, having really been reached in 

the same way, namely by multiplying the original expression by - . Cf. 

Appendix A, VII. I 5. These properties have been made use of by Nichol- 
son, see below V. ^ 3. 

On the same two suppositions the following formula (of Drobisch's 
method) would be serviceable, 

p r7 (^'2^2 + 2/2^2 -I- tow terms) 



— ^ (ccja^ -f 2/i/3j + to n terms) 



for, in the second, more comprehensive, case, this would reduce to 
p r7 (^2«2 + 2/2/^2 + to n terms) 



-^r7i^2°-i + y2f^i+ toTiterms) 2 1 \ yin i 



rn.; 



or to 






^' ^(a;iai + 2/A+ toTiterms) ^i"i + 2/i^i ^ 

Ill 

but not otherwise. 

Thus in cases when the mass-quantities are the same, or proportional, at 
both or all the periods compared, all the varieties of the method reviewed in 
this Section (and some of the methods to be reviewed in the next) are the 
same. ^ 

Now we know (see Chapt. XL Sect. III. §§2-3) that on the supposi- 
tions here made, namely of the mass-quantities being the same, or propor- 
tional, at both or all the periods compared, Scrope's method, applied to the 
same mass-quantities, or to the proportional ones at either period, or to any 
mean or average between them, is perfectly correct. This has been recog- 
nized, not only by Edgeworth and Nicholson, as above, but also by Drobisch, 
B. 29, pp. 30, 37, B. 30, pp. 147-148 ; Sidgwich, B. 56, pp. 66-67 ; Lehr, B. 
68, p. 40 ; Zuckerkandl, B. 115, p. 247, B. 116, pp. 244-245 ; Wicksell, B. 
139, pp. 8-9 ; Padan, B. 141. 

V. METHODS EMPLOYING DOUBLE WEIGHTING. 
§1- Drobisch's Method. Drobisch, 1871 ; see B. 29 and 30 (31 being 
apologetic ) . 

He always considered the variation of the arithmetic averages to be this, 

falXlk + ■'■■'■■ ' '"^^2/^henxai = 2//3,= , (B. 29, p. 36, cf. pp. 31, 39) 

— i. e., only when this expression is equivalent to - ( — + ^'^ + ) > (see 

2 This is not recognized by Powers, B. 131, pp. 27-28, where, by faulty arith- 
metic, he divides into two distinct methods what is really, in the case supposed, 
only one method. 



rp:view of methods 545 

Appendix A, II. §9). In other words, he always took " arithmetic average " 
(applied to price variations) to mean only arithmetic average loith even weight- 
ing. And then he rightly objected to the general use of the arithmetic aver- 
age so understood, B. 29, p. 32, and rightly condemned Laspeyres for using 
it, B. 30, p. 145, this average being correct only in the particular case when 
the mass-qujintities (numbers of the mass-units) happen to be in inverse pro- 
portion to the prices (of the same mass-units) at the first period. His objec- 
tion, evidently, was only against the use of even weighting in averagiiig price 
variations ; Init he thought it was against the use of the arithmetic average in 
general.^ He also objected to the method of using uneven weighting when 
only the mass-quantities at one of the periods are considered, the change in 
the mass-quantities at the other period being disregarded, and also to the 
method of using the mean between the two methods ( or the mean of the mass- 
quantities?), all these methods giving diflferent results, none better than 
another, B. 29, pp. 36-39, B. 31, pp. 423-425. He thought that "the fol- 
lowing consideration does away with all difficulties," B. 29, p. 39.- 

Taking his prices always to be the prices of the same mass-unit (preferably 
a hundredweight) of every class of goods, and consequently his mass-quanti- 
ties to be the numbers of these mass-units of all the classes of goods marketed 
at the place in question during the period in question, B. 29, p. 34, B. 30, p. 

148, he represented the total mass-quantities of the classes [A], [B], 

marketed at the first period hy x^ -\- y^ -\- , and the sums spent on them 

by XiOj^ 4" 2/i/?i -f- ; and hereupon he asserted that the average price of 

the mass-unit is no< - (^-~ -f ^ -f ), not - («, -|- A + ) , but 

n \ 2\ 2/i Jin J 

X CL —4-" 11 3 — T" ••••■• 

^ ' i —^^, ) (and remarked that even here he was not using the 

^i + 2/i4- 

arithmetic average, and concluded that his method does not use the arith- 
metic average at all, any more than it uses the harmonic or the geometric, 
but only the rule-of-three, B. 29, p. 40). Proceeding in the same way for the 
second period, he represented the then average price of a mass-unit of goods 

by " '' ~ ^'^ '^ . Therefore, neglecting any average of the price 

H + y-i-^ 

variations, he represented the variation of the price-average by the varia- 
tion of the average price of the common mass-unit of goods, thus, 

^■i _ -H ■¥ y-i ^ 

1*1 ^\^i + yA + 

2-1 + 2/1 -H 

^ a- 2»2 4- yA + ^1 + yi + 

•''i"! + Viftx + ' ^'2 + ^2 + ' 

1 He also noticed that if the mass-quantities happen to be in inverse propor- 
tions to the prices at the second period, the above expression reduces to the simple 
harmonic average of the price variations (of. Appendix A, IV. ?2) ; and con- 
cluded that even on the assumption of the mass-quantities being the same at 
both periods, the " arithmetic average " is no better than the "harmonic average," 
and vice versd, each being proper in certain circumstances and improper in all 
35 



546 APPENDIX C 

and the variation of the exchange-value of money (Oetdwert) , he added, is 
the inverse of this, B. 29, p. 39, B. 30, pp. 148-149. 

We see that this is nothing but a method using the arithmetic average 
with different numbers of figures at each period— i. e., with double weighting 
(cf. Appendix A, VII. | It- lli^ arithmetic average used by Drobisch, 
when he is drawing the average price of the mass-unit of all goods at each 
period separately, is the least imperfect distinct avei-age (for each period sep- 
arately) that he could draw, since it takes into account the numbers of times 
the mass-unit is repeated in every class, while the simple average, with even 
weighting of the classes, in the form rejected by him, would obviously 
be imperfect. But the fact that the average he rejected is an arithmetic 
average, does not prevent the average he adopted from being an arithmetic 
average.^ He thought he disproved the arithmetic and harmonic, and also 
the geometric, averages, and proved that a method using none of them is the 
right one. As a matter of fact, his arguments never touch upon the question 
as to which kind of average is the correct one, but only show that ail o^ 
them are wrong with even weighting — which everybody admits. And he re- 
adopted the arithmetic average without any argument, but assumed it. His 
attention was really concentrated upon the question of weighting, and it was 
here that he made advance. 

In this method it is material that the mass-unit should be the same in all 
the classes of things. For, although in the numerator for the average at each 
period every term (e. g., XiU.^) would be the same whatever mass-units are 
employed, yet in the denominator the corresponding term (e. g., x^) would 
be larger or smaller according as the mass-unit is smaller or larger.^ 

The peculiarity of Drobisch' s method, then, is that it uses the arithmetic 
average (1) with double weighting (2) on the same mass-unit for all the 
classes. 

This method is praised by Roscher, B. 32, p. 275 ; criticized by Laspeyres 
(see here Chapt. V. Sect. VI. l 5, Note 11), by Paasche (see here Chapt. 
IV. Sect. V. §1), and by Lehr and Zuckerkandl (see here Chapt. V. Sect. 
VI. § 5, Note 15) ; noticed by Edgeworth, B. 59, p.' 265 ; and reviewed by 
Lindsay (who said it rested on the harmonic mean!), B. 114, jjp. 18-21. 
It has even been used, in application to goods commonly measured in the 

others, B. 29, pp. 36-37 — a conclusion perfectly correct concerning these averages 
of the price variations, each with even-weighting, but without any application 
whatever against either with its proper weighting. 

2 Like Drobisch in condemning the "arithmetic average" while using it 
themselves (their objection being really against only the arithmetic average of 
the price variations with even weighting) are Geyer, B. 28, p. 321, and Paasche, 
B. 33, p. 171, cf. p. 172, B. 34, p. 60, cf. p. 64. Even Walras does not appear to 
perceive that the formula for Scrope's method (using the same mass quantities at 
each period) is one employing arithmetic averages (and is connected with the 
arithmetic average of price variations), sometimes contrasting it with the geo- 
metric, the harmonic and the arithmetic averages of the price variations (always 
with even weighting), as if these alone were averages. For the references, see 
above in IV. § 1 end. 

^ Drobisch's method is the only one yet invented in which this feature is 
essential. 



REVIEW OP" METHODS 547 

same mass-unit, without knowledge of Drobish, by Powers, B. 131, see p. 29. 

§ 2. Lehr's method. Lehr, 1885 ; see B. 68. 

Starting out, like Drobisch, by supposing the prices pf all articles to be ex- 
pressed on the same mass-unit (by weight), although the rest of his method 
makes this unnecessary, Lehr notices that over the two periods together there is 
spent the sum of (i,«i -f :)-2a2) money-units for {i\ + x^) mass-units of tlie class 
[A], wherefore the average price of one mass-unit of [A] for botli the periods 

is ' ^ , — —^ money-unit, and the money-unit i)urchases on the average dur- 

Xi + X.^ • ^ 

ing both the periods — ^^i——^ mass-unit of [A]. And similarly for the 

article [B] over the two periods together, there is spent the sum of 
(.Vi/^i + 2/2/^2) money -units for (2/1 + 1/2) mass-units, wherefore the average 

price of one mass-unit of [B] for both the periods is •^^— '— — ^^?^- ^ money-unit, 

2/1 + 2/2 
and the money-unit purchases on the average during both the periods 

-^ ~^- mass-unit. And so on with the other classes. Now in every 

]h(i^ + yA ... 

class the mass-quantity which the money-unit purchases on the average over 
the two periods together — or the mass-quantity of which the average price 
during the two periods together is one money-unit — he calls a "pleasure- 
unit," p. 38, having first defined a pleasure-unit to be an equivalent quantity 
of any goods, pp. 37-38. The total quantity of [A] which comes into trade 

during the first period consists of — -^ or j, ( ^ ^ — ^■^ 1 such pleas- 

•t'l + ^'2 \ ^1 + ^"2 / 

ure-units ; and the total quantity of [A] which comes into trade during tlie 

second period consists of a;.^ ( ^ ' _ i~~'~''' ) i^u^h pleasure-units. Similarly the 

total quantities of [B] coming into trade during each of the periods consist 

of yJy^^^y^A and y(yll±yS\ such pleasure-umts respectively. 

\ 2/1 + 2/2 / V 2/1 + 2/2 / 

And so on. During the first period, then, there come into trade 

AVh+j:2«2\ ^ (yA+Ml\j^ to n terms 

pleasure-units, for which are paid ( Xi«i + 2/1/^1 + ton terms) money- 
units ; wherefore the pleasure-units cost on the average, or their average price, 
expressed in money-units, is 



p _ ^i«i +2/1/^1 + 



/XiCi+xaX f yiPi + yA \ I 

And during the second period there come into trade 

/ a:,«,-j-.yA ^yA±yS)^ to n terms 

'V a^i+A-i / -"'V 2/1 + 2/2 / 



548 APPENDIX C 

pleasure-units, costing ( ■T2«2 + 2/2/^2 + tow terms) money -units ; where- 
fore 



^■2«2 + 2/2/^2 + 



-r-tTr)+-( 



2/1+2/2 / 



Therefore the variation of these average prices is the quotient of the expres- 
sion for P2 divided by the expression for Pj, which reduces to this, 



a'2«2 + 2/2/^2 + 



f x,a^ + x^a.A / 2/A + 2/ 2^2 \ , 



Pi ^iai + 2/A+ ,. /" ?i^2+^2«2 \ , „, fyA-\-y2p 



'=r-t^')+-(i^f)+ 



and the reciprocal of this, Lehr adds, gives the variation of the exchange- 
value of money, pp. 38-40. 

Comparing this formula with Drobisch's, we see that Lehr's formula differs 
from Drobisch's only in the second half, and there by his multiplying both 
Xi and X2 by a certain expression, which is the average price of the mass-unit 
of [A] during both the periods, and by multiplying both y^ and j/j by another 
expression, which is the average price of a mass-unit of [B] during both the 
periods, and so on, in every instance the average being the arithmetic aver- 
age with weighting according to the mass-quantities of each period. 

Hence the peculiarity of Lehr's method is that it uses the arithmetic aver- 
age (1) with double weighting (2) on mass-units that have the same (un- 
evenly weighted arithmetic) average price over both the periods compared. 

That the actual mass-unit used in every class is indifferent is plain from 
the fact that not only the terms, e. g., x^a^, in the first half, but also in the 

(T (t — (— 3' O \ 
^ ] ^ ''' ), have the same value whatever 
^"1 + % / 

be the mass-unit whose price is a and whose number is x. 

This method is reviewed by Edgeworth, B. 59, pp. 265-266 ; declared the 
best by Zuckerkandl, B. 115, p. 248, B. 116, p. 248, and Wiebe, B. 124, p. 
165 ; and by Lindsay, B. 114, pp. 22-24, criticised for seeking to measure 
variations in the mass-quantities ! Pointing out the error in this criticism, 
Wicksell tries to refute the method by a test case, Avhich supposes the prices of 
certain articles to be zero at the one or the other period [and which therefore 
constitutes a case in which these articles ought to be excluded from the 
measurement], B. 139, pp. 10-11. 

I 3. Nicholson's method. Nicholson, 1887 ; see B. 94. 

Independently, and seemingly without knowledge of Drobisch's method, 
or of Lehr's, Nicholson has invented a method which, being somewhat 
vaguely conceived, turns out, on analysis, to be either Drobisch's method mu- 
tilated, or an imperfect form of Scrope's, or something else. 

Kepresenting the aggregate wealth of a country expressed in the money- 
unit (the pound sterling) by i'Av, or by w, he sets out with a formula, for the 
first period, which in our notation is this, £,-iU\ = x^a^ -\- y-^(3-^ -{- ; where- 
fore the purchasing power [or exchange-value in all other things] of the 



REVIEW OF AfETHODS 549 

raonev-unit is ; j^- , pp. 307-308. Similarly at the second 

period, on the suppositit)n that the mass-quantities are about the same as 
before, and no new kinds of articles have been added or subtracted, we shovdd 
have £2«'2 =:a*2«2 + V'll^-i + > ^"<1 tlie purchasing power of the money-unit 

is ?i — , . Therefore we should have (representing the ex- 

a;2«2 + 2/2/^2 + 

change-value of money in all other things by Mo), 

1 

M02 _ x^a^ -\- yi(i.! -t- 

Moi ~ 1 

^i«i + 2/1/^1+ 

^2«2 + 2/2/^2 + ' 

This he represents as - , and says that, on these suppositions, "the 
change in the purchasing power of the £ is equal to the fraction ," and 

calls this fraction " the coefficient of appreciation," pp. 309-310. The same 
is true, he further says, if the mass-quantities have increased uniformly, 

which uniform increase he represents bv m, wherefore ??i = --'' = " = = 

— T ^' ; L on the ground that "the purchasing power of the stan- 

a;i4-2/i+ J 

dard will obviously be the same for the old inventory and for every 
uniform addition to it," so that, althougii the new coefficient of appre- 
ciation for these cases is - • m , yet either "the change in the standard may 
be measured by the old inventory at the old prices divided by the old in- 
inventory at the new prices )^. «., ^^ = — ^ ^^^^ + ...^TT J' PP" -^l"" 
311, or "the change in the purchasing power of the standard is found by 
dividing the value of the new inventory at the old prices by its value at the 

new" \i. e., ^^ -^'^ fi' l"' ^l P- ^^^- ^^^- '"^^"^^ ^^- ^ ^- 
Thus far we have merely Scrope's method inverted, because Nicholson is deal- 
ing with variations in the exchange- value of money in all other things.] All 
this he thinks accurate enough when we are dealing with periods near to- 
gether. His new method he begins when dealing with periods far apart, in 
which the classes of goods increase and decrease, and old ones become extinct 
and new ones come in— and all deserve to be allowed for. Here he again lets 
m represent the increase in the mass quantities. He is now very vague as to 
how m is to be estimated. He is willing to estimate it " generally," p. 312, 
or " from various independent considerations," p. 316. When the classes are 
the same at both the periods, he describes the estimation more minutely, in a 

%.f-^ -I- y-J^\ -\- Q1 9 Q1 Q 

way expressible in our notation thus, m= _-^ , ^ , , PP- f>^^ <5io 



550 APPENDIX C 

Ccf. Bourne in one of his methods, B. 47-49). When tlie classes are not all 
the same, he says "m should be proportioned to the increase in the popula- 
tion and to the increase in the efficiency of the labor and capital," p. 316. 

Such an increase would seem to be represented either by ^ . — ; 

^1 4- 2/1 +■•••■• 

(provided the numbers be expressed as those of some common mass-unit, 
about which Nicholson is silent), or, again supposing the classes to be the 
same at both the periods, the increase would seem to be represented by 

•j2 _j_ ^ _j_ \ _ However this bQ, the new coefficient of appreciation 






we find still to be — • m , which cannot be reduced now as it could be be- 

fore, p. 315. Hence, the variation of "genei'al prices" being the inverse of 
the purchasing power of money, pp. 317, 318, his formula is 

Pj Wjm ' 
which, in full, according to the methods of estimating m, is either 

P2 _ x^a^ + 2/2/32 + ^-i^i + 2/i/3i + 



Pj Xifli + 2/i/3i + x^aj^ + y^fii + 

P2 _ a;2«2 + 2/2/^2 + 2-1 + 2/1+ •••• 



(1) 
(2) 



Pi 2i«i + 2/1/^1 + ^2 + 2/2 + ' 

or 

P2^ a:2«2+ 2/2^2 + i/"^ I ^1 I \ ('i\ 

Pi %«i + 2/A + • n V a-2 "^ 2/2 "^ A ^' 

The first reduces to 

P2 ^ x^a^ + 2/2/^2 + 

Pi a;2ai + 2/2^^1 + ■■ ' 
which is Paasche's variety of Scrope's method. It is plainly no better than 



P2 


^2«2 +2/2/^2+ ^I«2 + 2/1/^2 H •■ 


pr 

ich reduces to 


•^■i«i + 2/1/^1 + a-ja., + 2/2/32 + 

P2 _ cCiOa + 2/A + 

Pi ■'Ki«i + 2/A + ' 



which is Laspeyres's variety of Scrope's method. Hence it is evident that in 
the second half some mean must be employed of the prices at the two periods. 

The second is the formula for Drobisch's method. But in this form Nich- 
olson's method is not the same as Drobisch's, since Nicholson does not specify 
how the mass-units are to be chosen. In this form this method is wholly 
haphazard. 

The third is the most original of the three forms.* But it contains a de- 

*A variation upon it may be suggested in the following, 



P2 ^ ^2«2 + 2/2/^2+ _ » /^l . 2/1 

Pi h^i + yil\ + '^ ^2 2/2 



REVIEW OF METHODS 551 

feet in averaging the inverted variations of the mass-quantities with even 

weighting. 5 

Which of these three the metliod really is, depends upon the way the 

ealculations have been made by the statisticians upon whom Nicholson is 

willing to rely. For a peculiarity in his method, as used by himself, is that 

lie makes separate use of the two halves in the formula. He accepts some 

statistician's rough appraisal of the comparative wealth, expressed in money, 

of a country at two periods, calculated, e. g., from the income tax, p. 318, 

... . . x,a, + y.,Bo + , , , . , 

(this giving 1~ o ~T > although no particular things are noticed, 

^1*^1 "T J/lPl + 

and permanent things, such as real estate and other capital, are included, if 
indeed the appraisal is not principally of them) ; and again he accepts some 
other statistician's rousjli estimate of the comparative wealth, in commodities, 
of the country at the two periods, calculated, e. </., from the increase of popu- 
lation and improvements in machinery and methods of production, ibid., (this 

giving, by inversion, either "-^ ; , , or - { - + --(- ), indef- 

^■2 + 2/2 + n\^2 2/2 / 

initely, without notice of particular things, but this time with reference 
chiefly to products). In this way he even thought he could show that be- 
tween 1848 and 1868 money appreciated, or " general prices " fell ! pp. 318- 
320. 

This method has been described by the Gold aud Silver Commission, 
Final Report, London 1888, p. 23, as "totally distinct" from the usual 
methods. Its partial similarity with Drobisch's method seems to have 
escaped notice. 

§ 4. Other methods involving double weighting we have found to be acci- 
dentally produced, in the comparison of later periods with one another, by 
methods using single weighting in the comparisons of the later periods with 
a common basic period. For these see above. III. | 4, and IV. § 2 (2). The 
latter of these may be described as a method the peculiarity of which is that 
it uses the arithmetic average (1) with double weighting (2) on mass-units 
that have the same price at some other period chosen as base. 

A method suggested in this work in Chapt. XII. Sect. II. § 4, Note 9, as 
an improvement on Lehr's is 

P2 ^ X2n^ + yi^'i+ •'- i(^i + «2^ + yi(/^i + /? 2) + 

P] ^l«l+2/l/?H ■^2(«l + «2) +2/1(^1 + /?2) + ' 

the peculiarity of which is that it uses the arithmetic average ( 1 ) with double 
weighting (2) on mass-units that have the same (evenly weighted arithmetic) 
average price over both the periods compared. 

^An improvement may therefore be suggested as follows, 

P2 _ ^2^2 4- 2/2/^2 H • • 1 ( „ '1 I 1, .'/i 

Pi ^i«i + 2/A+"- 
or better still, 

P2 arg gji 4- 2/2/^2 + S/ ( ^lY ( ViV^ + f 

Pi = ^«i-f2/A+-^-l^Vx-J KyJ ^«"^^""^' 

in which a, b, represent means between the weights of the two periods- 



-// ( a ■' ' + b •^' -f to *( terms ) 

n"\ A 2/2 / 



552 APPENDIX C/ / 

A method recommended in this work in Chapt. XII. Sect. II. is 

P2 ^ ^2«2 + 2/2/^2 + _ ^'i KliS + Vi V yiVi 4 

Pi a-ja^ + 2/i/3i + ^3 K^ia;2 + 2/2 1^2/12/2 + 

the peculiarity of which is that it uses the arithmetic average ( 1 ) with double 
weighting (2) on mass-units that have the same (evenly weighted geometric) 
average price over both the periods compared. 



'( 






BIBLIOGRAPHY 

OF WORKS DEALING WITH THE MEASUREMENT 

OF THE EXCHANGE-VALUE OF MONEY 

BY COMPARING MANY PRICES. 

W. Fleetwood 

1. Chronicon preciosum : or, an account . . . of the price of corn 
and other commodities . . . shewing from the decrease of 
the value of money . . . that a Fellow, who has an estate in 
land of inheritance, or a perpetual pension of five pounds per 
annum, may conscientiously keep his Fellowship, and ought 
not to be compelled to leave the same, tho' the Statutes of his 
College (founded between the years 1440 and 1460) did then 
vacate his Fellowship on such condition. — London 1707. (2d 
ed. 1745. See especially pp. 48-49, 135-137.) 

To lind the number of pounds which have the same exchange- 
value as the five pounds formerly liad, he inquires how much corn, 
meat, drink and cloth that sum would then pu^rfhase, and what sum 
is now needed to purchase tbem. [His proportions are so nearly 
the same, mostly six times, thai he escapes the question of aver- 
aging the present prices of the mass-quantities formerly priced at 
five pounds, and also the question of weighting. ] 

Dutot 

2. Reflexions politiques sur les finances et le commerce. — The 
Hague 1738, 18mo., Vol. I., pp. 365-377. 

Compares the total sums made up, at two periods (times of Louis 
Xn and Louis XIV), of the prices of the customary mass-units of 
various articles (including wages). [Uses the arithmetic average of 
prices with haphazard weighting. See Appendix C, I.] 

N. F. Dupr6 de Saint-Maur 

3. Essai sur les monnoies, on reflexions sur le rapport entre I'argent 
et les denrees.— Paris 1746, 4to, (The Reflexions occupy pp. 19- 
104. See also pp. 5-6, and in the second part pp. 161-182.) 

Uses many desultory price notices, but principally of grain, to 
compare the general prices of liis day with those in the period be- 
f(jre the discovery of America. His general conclusion is an average 
553 



554 BIBLIOGRAPHY 

of some sort, unspecified. But he uses the arithmetic average, with 
even weighting, in drawing the average price of single articles over 
many years. Criticizes Dutot for his data, not for his method. 

Gr. R. Carli 

4. Del valore e clella proporzione de' metalli monetati con i generi 
in Italia prima delle scoperte dell' Indie col confronto del valore 
e della proporzione de' tempi nostri. — 1764. (Ed. Custodi, 
Operescelte di Carli, Vol. I., pp. 299-366 ; in particular §IV., 
pp. 335-354.) 

Averaging the prices of grain, wine and oil, in the periods about 
1500 and 1750, he compares them by taking the earlier as units and 
reducing the later to the proper figures in proportion, thereby repre- 
senting the variations, and draws the arithmetic average, [thus using 
even weighting. See Appendix C, II.]. 

Gr. Shuckburgh Evelyn 

5. An account of some endeavors to ascertain a standard of weight 
and measure. (Philosophical Transactions of the Royal Society 
of London, 1798, Part I., Art. YIII., 4to., pp. 133-182. Only 
pp. 175-176, less than one page of printed matter, besides a 
table, devoted to the subject of measuring exchange-value, Hhe 
rest treating of weights and lengths.) 

Calculates the "depreciation of money" from 1050 to 1800 by 
taking the prices of 1550 at 100 and reducing the prices at the other 
dates in the same proportions. Each price-figure — for wheat, 
butcher's meat, day labo'- and twelve other agricultural products 
lumped together — is coi^,-ced as equally important, and the arithmetic 
average ist-draAvn. [Thus even weighting is used, but with subordi- 
nation of each of the twelve articles, i. e. irregular weighting by 
classification. See Appendix C, II., §3.] 

Arthur Young 

6. An enquiry into the progressive value of money in England as 
marked by the price of agricultural products : with observa- 
tions upon Sir G. Shuckburgh' s Table of Appreciation : the 
whole deduced from a great variety of authorities, not before 
collected. — London 1812, viii pp. and pp. 66-135. 

Objects mostly to Evelyn's prices and authorities, but also to his 
method, for counting every article as equally important. Counting 
wheat five times "on account of its importance" by value, barley 
and oats twice, provisions four times, day labor five times, wool, coal 
and iron each once, he multiplies the percentages of the variations 
betwen two periods by these weights, adds up the products, and di- 
vides by nineteen, the sum of the weights, [thus introducing uneven 
1 All this has been reprinted by R. Giffeu in the Bulletin de I'lnstitut interna- 
tional de Statistique, Eome 1887, pp. 132-134. 



BIBLIOGRAPHY 555 

weighting, with tlie arithmetic average of the price variations. See 
Appendix C, III.]. But in some calculations he also uses Evelyn's 
irregular weighting by classification. 

J. P. Smith 

7. The elements of the science of money. — Loudon 1813. (Ap- 
pendix, Art. I., " Estimate of the effective debasement of money 
in the eighteenth century," pp. 471-476.) 

Applies Evelyn's [form of Carli's] method (with strictly even 
weighting), to the reduced prices furnished by Young. [Very care- 
less.] 

Joseph Lowe 

8. The present state of England in regard to agriculture, trade, 
and finance. — London 1822, pp. 261-291, and Appendix, pp. 
85-101. 

Seeking to measure the "power of money in purchase" in order 
that debts may be paid in the same value, or at least that it may 
serve as a "table of reference," forming a "standard from ma- 
terials," he adopts Young's method, (but omits labor). 

Gr. Poulett Scrope 

9. Principles of political economy , . . applied to the present 
state of Britain.— London 1833, 18mo., pp. 405-408. 

Following Lowe in the object sought (the establishment of a 
"tabular standard " ), suggests this method : "Take a price-current, 
containing the prices of one hundred articles in general request, 
in quantities determined by the proportionate consumption of each 
article — and estimated ... in g(A:l. Any variation from time to 
time in the sum or the mean of these prices wiJ measure . . . the 
variations which liave occurred in the general exchangeable value of 
gold" (p. 406). [For the formulation of this see Appendix C, IV^.] 

[Henry James] 

10. The state of the nation. Causes and effects of the rise in value 
of property and commodities from the year 1790 to the present 
time. — London 1835, pp. 12-15 and a table. 

Measures the rise and fall in value [=; money-value, price] of 
Britisii produce, from 1798 to 1828, by the difference between the 
"official values" (always on the prices of 1694) and the "declared 
values" (according to current prices) of British exports. [Applies 
Scrope's method to custom-house returns. See Appendix C, IV., 
§2(2).] 

G. R. Porter 

11. The progress of the nation, in its various social and economical 
relations, from the beginning of the nineteenth century. — Lon- 
don 1838. (2d ed. 1847, pp. 439-440, 444-445.) 



556 BIBLIOGRAPHY 

Employs [Carli's] method on monthly prices of fifty articles in 
London from January 1833 to December 1837, the reduced price- 
figures being carried out to four decimal places. Calls the unit-price 
started with, the "index price." ^ 

M. C. Leber 

12. Essai sur 1' appreciation de la fortune privee au moyen age. — 
Paris, 2d ed. 1847, 335 pp. 

Eoughly measures the purchasing power of silver in relation to 
many commodities, etc., from the 8th and 11th centuries to the 
present, but differently for the poor and the rich. The method is 
not explained, [but is probably a rough arithmetic average of prices, 
reduced to francs] . 

R. H. Walsh 

13. Elementary treatise on metallic currency. — Dublin 1853, pp. 
94-96. 

Quotes and reviews, with apparent approval, Scrope's scheme and 
method. 

A. Soetbeer 

14. Das Gold. (Die Gegenwart, Leipzig 1856. Tables, pp. 587- 
588.) 

15. Ueber die Ermittelung zutrefFender Durchschnittspreise. 
(Vierteljahrschrift fiir Volkswissenschaft und Kulturgeschichte, 
Berlin 1864, Band IIL, pp. 8-32.) 

16. Materialien zur Erlaiiterung und Bem-teilung der wirtschaft- 
lichen Edelmetallverhaltnisse und der Wahrungsfrage. — Berlin 
1886, 4to., pp. 81-117. 

In the first gives tables of prices in Hamburg for 1831-40, 1841-50, 
1854, 1855, and of the same reduced to 100 at the first period ; but 
does not add or average them. In the second considers only the 
arithmetic average. In the third employs it on the prices, from 
1851 to 1885, on the basis 1847-50, of one hundred Hamburg and 
fourteen British articles, arranged in different divisions in which the 
articles are evenly weighted, and which, in the final averaging, are 
evenly weighted. [Evelyn's irregular weighting by classification.] 

J. Maclaren 

17. Sketch of the history of currency. — London 1858, pp. 311- 
312. 

Reviews Scrope's scheme and method. 

2 Porter's table wa,s put in evidence by J. B. Smith before the Select Commit- 
tee on Banks of Issue, 1840, p. 31, following q. 362. It was severely condemned, 
for the use of even weighting, before the same Committee, q. 3615, by Th. Tooke, 
who said that Porter had submitted to him the frame-work of his table before 
publication. 



BIBLIOGRAPHY 557 

E. Levasseur 

18. La question de I'or. — Paris 1858. (Measurements, pp. 179- 
195.) 

Uses partly [Dutot's] method, and partly [James's], the latter 
applied to similar French " official values" (at the prices of 1826) 
and "actual values," from 1847 to 185(3. [See Appendix C, IV. 
§2(2).] 

W. Newmarch 

19. Mercantile reports of the character and results of the trade of 
the United Kingdom during the year 1858 ; with reference to 
the progress of prices 1851-9. (Journal of the Statistical So- 
ciety of London, Vol. XXII., 1859, pp. 76-100.) 

20. Results of the trade of the United Kingdom during the year 
1859 ; with statements and observations relative to the cours-e 
of prices since the year 1844. (Ibid. Vol. XXIII., 1860, pp. 
76-110.) 

21. Commercial history and review of 1863. (Supplement to the 
Economist, Feb. 20th 1864, folio, pp. 4, 40-46.) 

In the first reduces the prices of twenty articles in the year 1851 
to 100, and gives their proportionate prices for the years following 
" ( omitting .^^, '54, '55, '56). In the second reduces the prices of 

twenty two articles in the period 1845-50 to 100, and gives their pro- 
portionate prices for the years following (with the same omissions). 
In the third continues the latter operations, and begins the series 
published annually in the Economist. In none are the reduced 
prices added or averaged. ( Tiie addition was first made in the Sup- 
plement of March loth 1869, where, on p. 44, appeared for the first 
time the " Total Index No." ) 

"W. S. Jevons 

22. A serious fall in the value of gold ascertained, and its social 
effects set forth. — London 1863. (Republished in Investigations 
in currency and finance , London 1884, pp. 15-118.) 

23. The variation of prices and the value of the currency since 
1782. (Journal of the Statistical Society of London, June 
1865. Republished in Investigations, pp. 119-150.) 

24. The depreciation of gold. (Letter in the Economist, May 
8th 1869. Republished in Investigations, pp. 151-159.) 

In the first raises the question whether the average of price varia- 
tions should be the arithmetic or the geometric, and adopts the latter, 
which he always uses with even weighting. The table contains 
thirty nine chief articles, from 1845 to 1862. In the second defends 
the geometric average against Laspeyres (see below No. 25), and 
introduces consideration also of the harmonic average. The table 
is extended to cover the years 1782-1865, on many articles in ten 



558 BIBLIOGRAPHY 

divisions [apparently with irregular weighting by classification, like 
Evelyn's]. 

E. Laspeyres 

25. Hamburger Waarenpreise 1850-1863 und die californisch- 
australischen Goldentdeckungeu seit 1848. Ein Beitrag zur 
Lehre von der Geldentwerthung. (Jahrbiicher fiir National- 
oekonomie und Statistik, Jena 1864, Band III., pp. 81-118.) 

26. Die Berechuung einer mittleren Waarenpreissteigerung. (Ibid. 
1871, Band XVI., pp. 296-314.) 

Employs [Carli's] method on the prices of forty eight articles, the 
basis being 1831-40, and coming down to 1863. Combats Jevons's 
geometric average and gives an argument for the arithmetic, in the 
first work. In the second defends his own position against Drobisch 
(see below No. 30) and Geyer (No. 28), against whose methods he 
delivers a counter-attack. Admits that uneven weighting ought to 
be employed, and gives an emended formula [for a method like 
Scrope's, but with the mass-quantities specified to be those of the 
first period — one of the formulae already rejected by Drobisch in No. 
29]. 

A. Walker 

27. The science of wealth. —Boston 1866. (5th ed. revised 1869, 
pp. 177-178, 481, 488.) 

Employs [Dutot's] method on the prices of ten articles in New 
York from 1834 to 1859, of seven in Calcutta from 1850 to 1854 and 
from 1863 to 1867, and of sixteen in Boston from 1862 to 1865. 

Ph. Geyer 

28. Theorie und Praxis des Zettel-Bankw^esens. — Munich 1867. 
(2d ed. 1874, Appendix, pp. 321-326.) 

Condemns Laspeyres's arithmetic and Jevons's harmonic (sic) 
averages, and then reconstructs a met!" " ch like Laspeyres's, 

differing only in reducing the first piicts lo about 2.00, [i. e. only 
nominally, and with slipshodness ; virtually Carli's method]. 

M. W. Drobisch 

29. Ueber Mittelgrossen und bie Anwendbarkeit derselben auf die 
Berechnung des Steigens und Sinkens des Geldwerths. (Be- 
richte iiber die Verhandlungen der Koniglich sachsischen 
Gesellschaft der Wissenschaften zu Leipzig ; Mathematisch- 
physische Classe. Band XXIII. , 1871, pp. 25-48.) 

30. Ueber die Berechnung der Veranderuugen der Waarenpreise 
und des Geldv^erthes. (Jahrbiicher fiir Nat.-oekon. und 
Statistik, 1871, Band XVI., pp. 143-156.) 

31. Ueber einige Einwiirfe gegen die ia diesen Jahrbiichern ver- 
offentlichte neue Methode, die Veranderuugen der Waaren- 



BIBLTOCJrvAPIIV -l")*) 

preise unci des Geldwerthes zu berechnen. {Ibid. 1871, Band 
XVI., pp. 416-427.) 

Would settle the dispute between Laspeyres and Jevons by re- 
jecting both the aritliinetic and the geometric averaging of price 
variations, on the ground that the mass-quantities ought to be 
taken into account at each period. Introduces a method employing 
double weighting, a feature in which is that the mass units are all 
the same. Thinks he rejects all averages, making use only of the 
rule-of-three. [But really compares the arithmetic average of the 
preciousness of commodities at each period. See Appendix C, V. 
§ 1, and Chapt. V. Sect. VI, § 5.] Himself formulates for the first 
time [Scrope's] method applied to the mass-quantities of the first, 
and of the second, period, and mentions a mean between the two. 

W. Eoscher 

32. Die Gruudlagen der Nationaloekonomie. — Stuttgart, 10th ed. 
1873, § 129, pp. 273-276. 

Approves of Drobisch's solution of tiie problem, (himself having 
appealed to Drobisch for the solution of it). 

H. Paasche 

33. Ueber die Preisentwickelung der letzten Jahre, nach den 
Hamburger Borsennotirungen. (Jahrbiicher fiir Nat.-oekou. 
und Statistik, 1874, Baud XXIII., pp. 168-178.) 

34. Studien iiber die Natur der Gpldentwerthung und ihre prak- 
tische Bedeutung in den letzten Jahrzehnten. — Jena 1878, 173 
pp. (Conrad's Sammlung nationaloekonomischer und statis- 
tischer Abliandluugeu, Band I.) 

Follows Drobisch in condemning both the geometric and the arith- 
metic averages, but, unlike Drobisch, wants the mass-quantities to 
be considered only of one period, preferably the later, [thus falling 
back on Scrope's method, like Laspeyres himself, except for the dif- 
ference in . '' iJje period at which the mass-quantities are 
chosen. See Appt^narx C, IV., §2 (2)]. Employs this method, in 
the first work, on twenty two articles on the basis 1847-67 through 
* the years 1868 to 1872, also with the average of these later years. 

[The table is continued by v. d. Borght, below No. 5-5, and Conrad, 
Nos. 96 and 97.] 

A. Hanauer 

35. Etudes economiques sur I'Alsace ancienne et moderne. Vol. 
II,, Denrees et salaires.— Paris 1878. (Chapt. XV., " Conclu- 
sion," giving a " resume general " of the "pouvoir de I'argent 
en general," pp. 601-609.) 

Employs unevenly weighted arithmetic average of the mass-quan- 
tities of ten articles purchasable with one franc (5 grammes silver at 
9/10) at quarter-century periods from 1351-75 down, compared with 



560 BIBLIOGRAPHY 

the mass-quantities purchasable 1851-75 as units. [Thus virtually 
uses the unevenly weighted harmonic average of price variations. ] 

A. Ellis 

36. The money value of food and raw materials. (The Statist, 
London June 8th 1878.) 

Applies [Young's] method to twenty five articles for the years 
1859, 1869, 1873, 1876 and the first quarter of 1878, the weighting 
being according to the itpportance of the articles in 1869, which is 
used as the base. 

Anonymous 

37. The rise in the value of gold. (Bankers' Magazine, London 
Oct. 1878, pp. 842-848.) 

Uses [Dutot's] method, comparing the prices of thirty one arti- 
cles in 1878 with their prices in 1868. 

R. GiflFen 

38. Report to the Secretary of the Board of Trade on the prices 
of exports of British and Irish produce in the years 1861-1877. 
(Parliamentary documents. Session 1879, c. 2247, folio, pp. 
2-15.) 

39. On the fall of prices of commodities in recent years. (Journal 
of the Statistical Society of London, March 1879 ; republished 
in Essays in Finance, London 1880, § I, the extent of the fall, 
pp. 312-322.) 

40. Report to the Secretary of the Board of Trade on the prices of 
exports of British and Irish produce, and the prices of imports, 
in the years 1861-78. (Parliamentary documents. Session 
1880, c. 2484, pp. 3-27.) 

41. Report to the Secretary of the Board of Trade on recent 
changes in the amount of the foreign trade of the United 
Kingdom and the prices of imports and exports. (Parliamen- 
tary documents, Session 1881, c. 3079, pp. 4-30.) 

42. Report to the Secretary of the Board of Trade on recent 
changes in the amount of the foreign trade of the United 
Kingdom and the prices of exports and imports. (Parliamen- 
tary documents. Session 1885, c. 4456, pp. iii-viii, 1-53.) 

43. Trade depression and low prices. (Contemporary Review, 
London June 1885 ; republished in Essays in Finance, Second 
Series, London 1886, §111., " The history of prices," pp. 16-22.) 

44. Index numbers. (Bulletin de I'Institut international de Sta- 
tisque, Rome 1887, 4to. pp. 126-131.) 

45. Evidence before the Royal Commission on Gold and Silver, 
First Report— London 1887, qq. 663-793, folio, pp. 34-42. 



BIHLIOGRAPHV 561 

Applies [Young's] method to custom-house returns, with the 
peculiarity that while his basis for prices in 1861, he takes the 
relative importance (according to total money-values) of the vari- 
ous classes in the year 1875 for the weighting through all the 
years, which are each compared directly with 18()1. [See Appen- 
dix C, III., ^3.] Tables at first, in No. 38, confined to exports 
for the alternate years from 1861 to 1877 ; later, in No. 40, applied 
also to imports, and extended to the succeeding years, and finally, in 
No. 42, carried backwards to 1840 for exports and to 1854 for im- 
ports. The primary object is to measure the volume of foreign trade. 

S. Bourne 

46. On some phases of the silver-question. (Journal of the Sta- 
tistical Society of London, June 1879 ; ou the fall of prices, 
pp. 413-417.) 

47. Ou the use of index numbers iu the investigation of trade sta- 
tistics. (British Association for the Advancement of Science, 
55th Meeting, 1885 ; published in the Keport, Loudon 1886, 
pp. 859-873.) 

48. Index numbers as illustrating the progressive exports of Brit- 
ish produce and manufactures. {Ibid. 58th Meeting, 1888 ; in 
Report, 1889, pp. 536-540.) 

49. Index numbers as applied to the statistics of imports and ex- 
ports. {Ibid. 59th Meeting, 1889 ; in the Report, 1890, pp. 
696-701.) 

In the first employs [Carli's] method, with strictly even weighting, 
but with attempt to improve upon the table of the I^conomist bv 
omitting some of the variations of cotton and adding coal, in a table 
from 1847 to 1879. In the others the object is rather to measure the 
volume of foreign trade, and the index numbers are of two kinds, 
referring to prices and referring to volumes. The former employ 
[Scrope's] method on the mass-quantities of 1883; the latter com- 
pare the mass-quantities which could have been bought for the sums 
actually spent at the prices of 1883 — both on sixty five articles in the 
custom-house returns for several years from 1865 down, [the latter 
sometiines involving the harmonic average of price variations]. 

A. de Foville 

50. La mouvement des prix dans le commerce exterieur de la 
France. (L' Economiste fran§ais, 1st article, July 5th 1879, 
folio, pp. 3-5 ; 2d, July 19th 1879, pp. 64-65 ; 3d, Nov. 1st 
1879, pp. 533-534. Second series, 1st article, April 29th 1882, 
pp. 503-505, 2d, June 17th 1882, p. 504.) 

51. Article " Prix " in the Nouveau Dictlonaire d'Economie 
politique. Paris 1892, Vol. II. (On price-measurement, pp. 
607-612.) 

36 



562 BIBLIOGRAPHY 

In the first applies [James's] method [as already done by Levas- 
seur] to the French "official " and " actual values" so long as pos- 
sible, from 1847 to 1862, and thereafter, down to 1880, a method by 
comparing the "provisional values" of each year (on the prices of 
the preceding year) with its "actual values." ^ [See in Appendix 
C, IV. (2). The object is rather to measure the volume of foreign 
trade]. The second is merely descriptive. 

A. Messedaglia 

52. II calcolo dei valori medii e le sue applicazioni statistiche. 
(Archivio di Statistica, Anno V., 1880. Republished by itself, 
Rome 1883, 86 pp.; on prices, pp. 36-40.) 

Examines the raathematics of the three means, and, pointing out 
the inverse relationship between the arithmetic and the harmonic, 
thinks that for measuring the purchasing power of money over goods 
we want the arithmetic average of the variations of the mass-quanti- 
ties, and therefore the harmonic average of the price variations, or 
for measuring the power of goods over money, directly the arithmetic 
average of the price variations ; but for the geometric average there 
is no place. Pays little attention to weighting. 

H. 0. Burchard 

53. lu the Fiuance Reports of the Secretary of the Treasury, 
Washington, for the years 1881 pp. 312-321, 1882 pp. 252-254, 
1883 pp. 316-318 ; and in the Report of the Director of the 
Mint on the production of the precious metals in the United 
States, for the year 1884, pp. 497-502. 

Uses [Carli's] method. The tables give average prices for 1883 
and 1884 compared with 1870, and with 1882, and also with the 
general average for fifty six years ending 1880. 

G. Fauveau 

54. Comparaison du pouvoir de la mounaie a deux epoques dif- 
ferents. (Journal des Economistes, Paris June 1881, pp. 354- 
359.) 

Formulates [ScrojDe's] method. 

R. V. d. Borght 

55. Die Preisentwickelung wahrend der letzten Decennien nach 
der Hamburger Boi-sennotirungen. (Jahi-biicher fiir Nat. - 
oekon. und Statistik, 1882, N. F. Band V., pp. 177-185.) 

Continues Paasche's tables [see above No. 33] down to 1880, using 
the same method. 

H. Sidgwick 

56. Principles of political economy. — London 1883. (Book I.) 

3 This method has been continued in several of the Bulletins du Ministere des 
Finances. 



BIBLIOGRAPHY 563 

Chapt. II., " On the definition and measure of value," pp. 52- 
69.) 

Eecommends [Scrope's] method, with the mass-quantities of the 
first or of the second period, or of a mean [the arithmetic] between 
the two. None of tliese being better tlian anotlier, tliere is no 
single authoritative measurement. 

P. Y. Edgeworth 

57. On the method of ascertaining a change in the value of gold. 
(Journal of the Statistical Society of London, Dec. 1883, Vol. 
XLVI., pp. 714-718.) 

58. The choice of means. (The London, Edinburgh and Dublin 
Philosophical Magazine and Journal of Science, Sept. 1887, pp. 
268-271.) 

59. Memorandum of the Secretary, attached to the First Report 
of the Committee of the British Association [see below No. 
99] , at the 57th Meeting, 1887. (Published in the Report of 
that Meeting, London 1888, pp. 254-301.) 

60. Memorandum on the accuracy of the proposed calculations of 
index numbers, attached to the Second Report of the same 
Committee [see below No. 100], at the 58th Meeting, 1888. 
(Published in the Report, 1889, pp. 188-219.) 

61. Some new methods of measuring variations in general prices. 
(Journal of the Royal Statistical Society, London, June 1888, 
pp. 346-368.) 

62. Appreciation of gold. (Quarterly Journal of Economics, Bos- 
ton Jan. 1889, pp. 151-169.) 

63. Memorandum, attached to the Third Report, of the same 
Committee as above [see below No. 101] , at the 59th Meeting, 
1889. (Published in the Report, 1890, pp. 133-164.) 

64. Recent writings on index numbers. (Economic Journal, Lon- 
don 1894, pp. 158-165.) 

65. Articles "Average" and "Index Numbers" in Palgrave's 
Dictionary of Political Economy, Vol. I., London 1894, p. 74, 
Vol. II:, 1896, pp. 384-387. 

66. A defence of index numbers. (Economic Journal, March 
1896, pp., 132-142.) 

Treats of the measurement of the variations of the value of money 
under many [often artificial] conceptions of the qucesitum. Recom- 
mends generally the aritlimetic average with even or uneven weight- 
ing, but also the median, and again the geometric ( this principally 
with even weighting), according to the object sought. 

L. Hansard 

67. On the prices of some commodities <luring the decade 1874-83. 



U 



564 BIBLIOGRAPHY 

(Paper read before the Bankers' Institute, London Dec. 17tli 
1884, and published in their Journal, Jan. 1885, pp. 1-42.) 

Employs only addition of the prices reduced on a common scale 
[like the Economist at first, virtually Carli's method]. 

J. Lehr 

68. Beitrage zur Statistik der Preise, insbesondere des Geldes und 
des Holzes. — Frankfurt a. M. 1885. (On the method, pp. 11, 
37-42.) 

Follow' s Drobisch in employing double weighting, but seeks a 
common unit for both the periods compared, which he calls a 
"pleasure-unit," and which he finds in the mass-quantity of every 
class whose unevenly weighted arithmetic average price over both 
the periods is one money-unit. [See Appendix C, V. § 2.] 

L. Walras 

69. D'une methode de r6gularisation de la valeur de la monnaie. 
(Memoire read before the Societe vaudoise des Sciences natur- 
elles, June 6th 1885 ; published, Lausanne 1885, 22 pp. ; also 
republished below in No. 71.) 

70. Elements d' economic politique pure. — 2d ed., Lausanne 1889. 
(Pp. 431-432, 457-468, partly incorporating the preceding.) 

71. ;6tudes d'economie politique appliquee. — Lausanne and Paris 
1898. (Mostly reprints, with some new matter.) 

Follows Jevons in adopting the geometric average of price varia- 
tions with even weighting, and compares it with the arithmetic and 
harmonic averages, also only with even weighting, all which he for- 
mulates. Later inclines to prefer what he calls "the formula of the 
multiple standard" [Scrope's method], because it takes into account 
the mass-quantities [of which period, he does not consider]. 

A. Simon and L. Walras 

72. Contribution a 1' etude des variations des prix depuis la sus- 
pension de la frappe des ecus d' argent. (Memoire read before 
the Societe vaudoise des Sciences naturelles, June 3d 1885, 
published jointly with No. 69, Lausanne 1885, 11 pp.; also re- 
published in No. 71.) 

Apply the geometric average with even weighting to the prices 
of twenty articles at Berne from 1871 to 1384 on the basis of the 
arithmetic average of prices during the period 1871-78. 

[C. D. Wright] 

73. Sixth annual Report of the Bureau of Statistics of Labor. — 
Boston 1885, pp 154-156. 

Compares prices in Great Britain and in Massachusetts, for work- 
ing men, using [Scrope's] method twice applied, on the mass-quan- 
tities of each country. 



BIBLIOdRAPHY 565 

M. G. Mulhall 

7-i. On the variations of price-levels since 1850. (British Assoc- 
iation, 55th Meeting, 1885, epitomized in the Report, 1886, pp. 
1157-1158.) 

75. History of prices since the year 1850. — Loudon 1885, 190 pp. 

Employs [Scrope's] method, with the mass-quantities of the later 
periods [in Paasche's form]. Claims to apply it to the whole 
world. Calls it "the volume of trade method." 

S. Newcomb 

76. Principles of political economy. — New York 1886. (Book 
III. Chapt. II., "The measure of value by an absolute stan- 
dard," pp. 205-214.) 

Recommends [Scrope's] method, 

R. H. Inglis Palgrave 

77. Currency and standard of value in England, France, and India, 
and the rates of exchange between these countries. (Memor- 
andum laid before the Royal Commission on Depression of 
Trade and Industry, 1886, Third Report, Appendix B, folio, 
pp. 312-390.) 

Gives various tables of index-numbers, correcting the Econo- 
mist's figures for England by weighting the price variations accord- 
ing to the money-values at the later periods [Young's method, more 
specified as to the weighting, see Appendix C, III. § 4], also for 
India and France, mostly from 1865 to 1886. 

F. B. Forbes 

78. The causes of depression in the cotton industry of the United 
Kingdom. (Occasional Paper of the Bimetallic League, No. 
3.) London August 1886, pp. 12, 18, 20. 

Applies Jevons's simple geometric mean to Barbour's figures of 
quantities per rupee, comparing 1884-85 with 1875-76 on twelve 
classes of exported, and seven of imported, goods. 

A. Sauerbeck 

79. Prices of commodities and the precious metals. (Journal of 
the Statistical Society of London, Sept. 1886, pp. 581-631, 
Appendix, pp. 632-648.) 

80. Prices of commodities in 1888 and 1889. {Ibid. March 1890, 
pp. 141-135.) 

81. Prices of commodities in 1890. {Ibid. March 1891, pp. 128- 
137.) 

82. Prices of commodities during the last seven years. {Ibid. 
June 1893, pp. 215-238, Appendix, pp. 239-247 and 254.) 

83. Prices of commodities in 1893. {Ibid. March 1894, pp. 172- 
183.) 



566 BIBLIOGRAPHY 

84. Prices of commodities in 1894. (Ibid, March 1895, pp. 140- 
154.) 

85. Index numbei-s of prices. (Economic Journal, June 1895, pp. 
161-174.) 

86. Prices of commodities in 1895. (Journal of the Statistical So- 
ciety, March 1896, pp. 186-201.) 

87. Prices of commodities in 1896. (Ibid. March 1897, pp. 180- 
194.) 

88. Prices of commodities in 1897. (Ibid. March 1898, pp. 149- 
162.) 

89. Prices of commodities in 1898. {Ibid. March 1899, pp. 179- 
193.) 

90. Prices of commodities in 1899. {Ibid. March 1900, pp. 92- 
106.) 

Applies [Carli's] method to the prices of forty five articles on the 
bases of prices in 1867-77, going back to 1848 and continuing to the 
present. Also adds (beginning in No. 82) two "tests," which seem 
to be [Young's] method, on the relative money-values in 1889-91, 
and [Scrope's] method on the mass-quantities of the other years 
[like Paasche's]. In No. 86 (pp. 193-194) he experiments with the 
geometric mean. 

F. Coggeshall 

91. The arithmetic, geometric, and harmonic means. (Quarterly 
Journal of Economics, Oct. 1886, pp. 83-86.) 

Discusses the three means, mostly from the point of view of avoid- 
ing error in the result arising from errors in the data, without attach- 
• ing superiority to any one of them, regarding the mean of prices as a 

' ' fictitious mean. ' ' 

J. E. Thorold Rogers 

92. A history of agriculture and prices in England, Vol. V. — Ox- 
ford 1887. (Chapt. XXVI., "On prices generally between 
1583 and 1702," pp. 778-800.) 

Employs [Carli's] method. 

A. Marshall 

93. Remedies for fluctuations of prices. (Contemporary Review, 
London March 1887, Sect. V., " How to estimate a unit of pur- 
chasing power," pp. 371-375.) 

Assumes the arithmetic average. 

J. S. Nicholson 

94. The measurement of variations in the value of the monetary 
standard. (Paper read before the Royal Society of Edinburgh, 
March 21st 1887 ; published in the Journal of the Statistical 



BIBLIOGRAPHY 567 

Society of London, March 1887, and republished in Treatise on 
money, Edinburgli and London 1888, pp. 298-331.) 

Invents a new method [which is partly a variation upon Drobisch's. 
See Appendix C, V. § 3J. 

A. Beaujon 

95. Sur la question des " index numbers." — Propositions soumises 
^ I'Institut international de Statistique en vue d'obtenir des 
tableaux de prix moyens comme base du calcul des index num- 
bers. — Index numbers ou chiffres de prix de marchandisesdans 
divers etats, depuis 1870. (Bulletin de- I'Institut international 
de Statistique, 1887, pp. 106-11-1, 115-116, 117-126.) 

Discusses manner of collecting data. Leaves the question of aver- 
ages to a future deliberation of the Institute.'' In the third reports 
several tables of the writers above. 

J. Conrad 

96. Beitrage zur Beurteilung der Preisreduktion in den 80 er 
Jahren. (Jahrbiicher fiir Kat.-oekon. und Statistik, 1887, N. 
F. Band XV., pp. 322-331.) 

97. Die Entwickelung des Pi-eisuiveaus in den letzten Decennien 
und der deutsche Getreidebedarf in den letzten Jahren. {Ibid. 
1899, DritteF. BandXVIL, pp. 642-660.) 

Continues Paasche's and v. d. Borght's tables down to 1885, and 
later to 1897, using the same method. 

F. Krai 

98. Geldwertund Preisbewegung im Deutsehen Reiche 1871-1884. 
— Jena 1887. (On prices, pp. 63-111.) 

Uses the arithmetic average with both [Dutot's] haphazard weight- 
ing and [Evelyn's] weighting by classification. 

British Association for the Advancement of Science : 

Committee consisting of S. Bourne, F. Y. Edgeworth, H. S. Fox- 
well, R. Giffen, A. Marshall, J. B. Martin, J. S. Nicholson, R. 
H. I. Palgrave and H. Sidgwick, appointed for the purpose of 
investigating the best methods of ascertaining and measuring 
variations in the value of the monetary standard. 

99. First report, to the 57th Meeting, 1887. (Published in the 
Report, Loudon 1888, pp. 247-254.) 

*A Comite de la Statistique des Prix Avas appointed by the Institute, consisting 
of Beaujon, de Foville, de luama-Sternegg, Gilfen, de Neumann-Spallart, de Mayr, 
and Pantaleoni. But beside brief reports on recent works, made by Martin and 
Palgrave, October 1891 and September 1893 (in the Bulletin, 1892, pp. 24.5-246, 
and 1895, pp. 57-60), this Committee does not appear to have made an original 
report. 



568 BIBLIOGRAPHY 

100. Second report, to the 58tli Meeting, 1888. {Ibid. 1889, pp. 
181-188.) 

101. Third report, to the 59th Meeting, 1889. {Ibid. 1890, p. 133.) 

102. Fourth report, to the 60th Meeting, 1890, {Ibid. 1891, pp. 
485-488.) 

Review several forms of [Scrope's] method, and discuss various 
questions connected with price-measurements. 

E. Nasse 

103. Das Sinken der Warenpreise wahrend der letzten fiinfzehn 
Jahre. (Jahrblicher fiir Nat.-oekon. und Statistik, 1888, N. F. 
Band XVII.; on methods of measurement, pp. 51-53.) 

104. Das Geld und Miinzwesen. (Schdnbergs Handbuch der Po- 
litischen Qekonomie, Tlibingen, Vol. II., 1890; on methods of 
measurement, pp. 331-332, folio.) 

Reviews some of the methods using the arithmetic average. 

K. Wasserab 

. 105. Preise und Krisen. Gekronte Preisschrift iiber die Veran- 
derungen der Preise auf dem allgemeinen Markt seit 1875 und 
deren Ursachen. — Stuttgart 1889. (On price-measurements, 
pp. 75-128.) 

Applies a method of his own [which is really Young's] to prices 
on the basis 1861-70 down to 1885. 

F. Schmid 

106. Bericht liber die Thatigkeit des statistischen Seminars an der 
k. k. Universitat Wien im Wintersemester 1888-89. (Statis- 
tische Monatschrift, XV Jahrgang, Vienna 1889. On variations 
in the purchasing power of money, aided by G. H. Thierl, pp. 
643-650 ) 

Reviews several methods. 

E. B. Andrews 

107. An honest dollar. (Publications of the American Economic 
Associatipn, New York, Vol. IV., No. 6, Nov. 1889 ; see pp. 
38-39.) 

108. Institutes of economics. — Boston 1891, pp. 141-142. 

Recommends [Scrope's] method, but allows the use of the geo- 
metric, arithmetic, or harmonic means. 

K. T. von Inama-Sternegg 

109. Der Riickgang der Waarenpreise und die oesterreichisch- 
ungarische Handelsbilanz 1875-1888. (Statistische Monat- 
schrift, XVI Jahrgang, 1890. Tables, pp. 6-7.) 



BIBLIOGRAPHY 509 

Gives tables applying [Carli's] method (in a form like the Econo- 
mist's) to thirty articles of import and to twenty-five of export in 
Austria-Hungary from 1880 to 1888 on the basis of 1875-1879. 

H. Westergaard 

110. Die (Trundziige der Theorie der Statistik.— Jena 1890. (Ou 
price-measurements, pp. 218-220.) 

Points out that the geometric average [he supposes the same 
weighting throughout, having even weighting mostly in miud] gives 
the same index-numbers in a series of periods whether applied to 
comparing each subsequent period with the original base or to com- 
paring any of the subsequent periods with each other ; and that this 
is not done by the usual methods employing the arithmetic average 
[Carli's and Young's]. Oflers this as an argument for the geometric 
average. 

R. P. Falkner 

111. Report of the Statistician. (Pp. xi-c in Vol. I. of the Re- 
port on retail prices of Mr. Aldrich from the Committee on 
Finance, 52d Congress, 1st Session, No. 086. Washington 
1892.) 

112. Report of the Statistician. (Pp. 27-337 in Vol. I. of the Re- 
port on wholesale prices of Mr. Aldrich from the Committee 
on Finance, 52d Congress, 2d Session, No. 1394. Washington 
1893.) 

113. Wholesale prices: 1890 to 1899. (Bulletin of the Depart- 
ment of Labor, No. 27, Washington March 1900 ; pp. 237-313.) 

Applies both [Carli's] and [Young's] methods to prices in the 
United States, in the first comparing 1891 with 1889, in the second 
extending the investigation to the years 1840-1891 on the basis of 
1860, and in the third bringing it down to 1899. 

S. McC. Lindsay 

114. Die Berechnung der Edelmetalle seit 1850. —Jena 1893. 
(Conrad's Sammlung. On the method, pp. 9-28.) 

Reviews several methods, and adopts [Scrope's] with the mass- 
quantities of the first or of the second period, or a mean between 
them, according as any of these best represents the importance of 
the classes. 

R. Zuckerkandl 

115. Die statistische Bestimmung des Preisuiveaus. (Handwor- 
terbuchder Staatswissenschaften Jena, Vol. V., 1893, 4to., pp. 
242-251.) 

116. La mesure des transformations de la valeur de la monnaie. 
(Revue d'economie politique, Paris 1894, pp. 237-253.) 



570 BIBLIOGRAPHY 

Says that [Scrope's] method is correct when the mass-quantities 
are the same at both periods; when they are different, Lehr's method 
comes the nearest to the truth. — For the purpose of paying contracts 
with money of stable purchasing power, recommends another method, 
which is not clearly described, [but which seems to revert to Scrope's, 
applied to the mass-quantities at the time of making the contract]. 
(The second is a translation of the first, Avith a few changes. Each 
contains a brief bibliography. ) 

G. d'Avenel 

1 1 7. Histoire 6conomique de la propriete, des salaires, des denr6es, 
et de tous les prix en general depuis I'an 1200 jusqu'en I'an 
1800.— Paris, 4 vols., 4to., 1894-1898. (On the method. Vol. 
I., pp. 6-13; results, pp. 27, 32, 137.) 

118. La fortune privee a travers sept siecles. — Paris 1895, 16mo. 
(On the methods, pp. 4-14 ; results, p. 37.) 

Apparently draws, very roughly, an average (the arithmetic) of 
the mass-quantities of goods purchasable with given amounts of silver 
at different epochs from 1200 to 1890, [thus using the harmonic 
average of their prices], with uneven weighting (in budgets of the 
expenditures of three different classes of society). (The second an 
abridgement of the first.) 

T. H. Whitehead 

119. The critical position of British trade with Oriental countries. 
(Paper read before the Eoyal Colonial Institute, Feb. 12th 1895, 
and reprinted from the Proceedings of the Institute. Table on 
p. 35.) 

Gives index-numbers compiled by W. S. Wetmore, on the plan of 
the Economist [Carli's method], for twenty articles in China. 

J. A. Fraser and C. H. Sergei 

120. Sound money. — Chicago 1895. (A measurement, p. 171.) 

Casually apply [Dutot's] method to four articles, comparing 1871 
and 1891. 

F. W. Taussig 

121. Results of recent investigations on prices in the United States. 
(Bulletin de I'Institut international de Statisque, 1895, pp. 22- 
32.) 

Reviews Falkner's works (Nos. Ill and 112) and discusses some 
points in connection with index-numbers. 

N. Gr. Pierson 

122. Index numbers and the appreciation of gold. (Economic 
Journal, September 1895, pp. 329-335.) 



BIBLIOGRAPHY ^71 

Is content with the arithmetic average. Prefers Soetbeer's results 
because of the great number of articles used.^ 

A. L. Bowley 

123 Comparison of the rates of increase of wages in the United 
States and in Great Britain 1860-1891. (Paper read before the 
British Association, Sept. 12th 1895 ; published in the Economic 
Journal, 1895; price-measurements, p. 381.) 
Uses the arithmetic average. 

G. Wiebe 

124. Zur Geschichte der Preisre volution des XVI und XVII Jahr- 
hunderts.— Leipzig 1895. (On the method, pp. 163-174; 
tables, pp. 369-386.) 

Applies both [Carli's] and [Evelyn's] methods to European prices 
from 1451 to 1700. Recommends Lehr's method for present re- 
searches. 

H. Denis 

125. La depression 6conomique et sociale et I'histoire des prix.— 
Ixelles-Bruxelles 1895. (On price-measurements, pp. 9-34.) 

Uses the arithmetic average with even weighting [Carli's method]. 
This weighting he considers sufficiently approximative in practice, 
though wrong in theory. 

G. M. Boissevain 

126. La question mon^taire. (M6moire traduit du Hollandais.) 
—Paris 1895. (On the method, pp. 52-53 ; and two tables.) 

Modifies Sauerbeck's index-numbers by calculations upon British 
exports and imports. 

A. L. Fonda 

127. Honest money. -New York 1895, 12 mo. (On "the stan- 
dard of value," pp. 158-161, 165.) 

For his standard would use [Young's] method, to be applied to 
a hundred staple commodities. 

Wharton Barker 

128. The course of prices. (The American, Jan. 25th 1896, foho, 
pp. 54-56, and quarterly since.) 

Gives index-numbers for American prices on a hundred and one 

articles in thirteen groups from Jan. 1st 1891, quarterly, continued 

to the present, using the arithmetic average of the price variations 

with even weighting, all compared with the first period (following 

the example of the Economist in England). 

6 In Further considerations on index-numhers, in the same Journal March 

1896, pp. 127-131, he rejects the whole system of index-numbers because diflferent 

results can be obtained on the same price variations [by using different weights]. 

(Reply by Edgeworth, No. 66.) 



572 BIBLIOGRAPHY 

J. Allen Smith 

129. The multiple money standard. (Publications of the Ameri- 
can Academy of Political and Social Science. Philadelphia 
1896. On the measurement of the standard, pp. 27-30.) 

For his standard would use [Scrope's] method, the mass-quantities 
chosen to be revised from time to time (at long intervals), all com- 
modities being included that can be accurately defined as to quantity 
and quality. 

T. N. Whitelaw 

130. A contribution to the study of a constant standard and just 
measure of value. — Glasgow 1896. (On the standard, pp. 18- 
19, 32-35.) 

For his standard would use [Carli's] method, to.be applied to 
about twenty of the chief agricultural products. 

L. Gr. Powers 

131. Fifth annual report of the Bureau of Labor of the State of 
Minnesota, 1895-1896.— St. Paul 1896, 524 pp (On the 
methods used, pp. 26-30.) 

Uses (1) [Drobisch's] method in groups of articles whose prices 
are reported in the same mass-unit, (2) [Scrope's] method applied to 
the arithmetic average of the mass-quantities over thirty five years, 
(3) Sauerbeck's "corrected method" [Paasche's, or Scrope's ap- 
plied to the mass-quantities of the later years singly], and (4) the 
' ' simple ' ' arithmetic average of the price variations [Carli' s method] . 
Calculations confined to agricultural products, principally in the 
West, extending from 1862 to 1895, on the basis of 1872. [Are 
vitiated by combining the prices of widely separated localities, and 
by attempting to eliminate the effects of reduced costs of transporta- 
tion. ] 

M. Bourguin 

132. La mesure de la valeur et la monnaie. — Paris 1896, 273 pp. 
(On index numbers, pp. 134-139.) 

Although denying the existence of general exchange-value, wants 
to measure the average of the variations of all the particular ex- 
change-values of money. Approves of the aritlimetic average, with 
even weighting. 

L. L, Price 

133. Money and its relations to prices — being an enquiry into the 
causes, measurement, and effects of changes in general prices. ■ 
— London 1896, 12 mo. (On the measurement, pp. 9-36.) 

Briefly surveys the subject. 

F. J. Atkinson 

134. Silver prices in India. (Journal of the Royal Statistical 
Society, March 1897, pp. 84-147.) 



BIBLIOGRAPHY 573 

Applies [Young's] method to one hundred articles in forty groups 
in various parts of India from 1861 to 1895, on the price-basis of 
1871, with weighting according to the total money-values of the 
groups in 1893-94. 

Eltweed Pomeroy 

135. The multiple standard for money. (Arena, Boston Sept. 
1897. On the method, i^p. 331-333.) 

For his standard would use [Scrope's] method applied to ihe mass- 
quantities consumed by working men's families, employing two hun- 
dred staple articles, their prices being collected from one hundred 
centers of commerce. 

F. Parsons 

136. Kational money. A national currency intelligently regulated 
in reference to the multiple standard. — Philadelphia 1898. (On 
the standard, pp. 113-138.) 

For his standard would use [Scrope's] method applied to a 
couple of hundred articles in the mass-quantities that are consumed 
by the average- family, the list to be revised from time to time 
(frequently). 

R. Mayo-Smith 

137. Movements of prices. (Political Science Quarterly, New 
York Sept. 1898, pp. 477-494.) 

138. Statistics and economics. — New York 1899. (Chapt. VI., 
"Prices"; on the measurement, pp. 199-228.) 

Briefly reviews the problems connected with index-numbers. (The 
second slightly expanded from the first. ) 

K. Wicksell 

139. Geldzins nnd Giiterpreise. — Jena 1898. (On price-measure- 
ments, pp. 6-16.) 

Recommends [Scrope's] method, provided the results are the 
same on the mass-quantities of each period. Otherwise the problem 
is unsolvable, as the measurement with the mass-quantities of the 
one period is as good as with those of the other, and the mean be- 
tween the two can have only "conventional meaning." 

A. W. Flux 

140. Some old trade records re-examined : a study in price-move- 
ments during the present century. (Paper read Feb. 8th 
1899 before the Manchester Statistical Society, and published 
in their Transactions, Session 1898-99, pp. 65-91.) 

Applies [.James's] method to British prices 1798-1869, to French 
prices 1873-97 [cf. De Foville, No. 50], and to German prices 
1891-97. 



574 BIBLIOGRAPHY 

E. S. Padan 

141. Prices and index numbers. (Journal of Political Economy, 
Chicago March 1900, pp. 171-202.) 

Considers the arithmetic average the only rigorous one. Attacks 
Jevons for "beclouding" the subject by introducing the geometric 
average and suggesting the harmonic. Advocates recognition of 
mass-quantity. This being done, the method [Scrope's] he thinks 
to be accurate provided the mass-quantities are proportional at all 
the periods compared, because then the same results are obtained 
of whatever period the mass-units be used. But not so, if the mass- 
quantities are irregular, so that in such cases no one result is authori- 
tative. 



INDEX 



[Simple numerals refer to par/es. Follouyin.r/ the letter B they refer to ivorks in 
the Bihliogriiphii.l 



Absolute : of value 3n. ; of variations 
57-8, 66 ; its meanings 66-7. 

Absoluteness, want of, no defect in 
mensuration 65-6. 

Aggregate of exchange-values, varia- 
tions of 439, 459 ; measurement of 
its variations 459-61. 

Appearance of new classes 112-14. 

Aldrich Report 475-6 ; see B. 111-12. 

Andrews 480n., 540 ; B, 107, 108. 

Api)roacli : correction bv 103-4, 105, 
117, 118, 420-3. 

Aristotle 123n., 137n. 

Arithmetic average and mean : pref- 
erence for 218-19, 222-3 ; careless 
rejection of 223, 546n. ; argument 
for, in general, refuted 249-51, 
253-4 ; argument for, in averaging 
prices 229,257-8 ; disproved when 
the sums are constant 296-8 ; study 
of 502-6. 

Arrivabene 438n. 

Atkinson, F. J., B. 134. 

Avenel, G. d' 219n., 477n.; B. 117- 
18. 

Average : led to by the formuloe 145 ; 
suggested 216 ; comparison of, and 
average of variations 157, 159 (cf 
164), 184, 185, Appendix A.; must 
be constructed 158-9, 185 ; ques- 
tion of, raised by Jevons 220, 223 ; 
the geometric, to be distinguished 
from the geometric mean 239, 
241-3, 255, 436 ; not so with the 
arithmetic and harmonic 239-40 ; 
nature of, in general 498-501. 

Bailey 23n. 

Bankers' Magazine 534; B. 37. 

Barbour 219n. 

Barker 536 ; B. 128. 

Bascom 438 n. 

Basic period, unimportance of 208. 

Beaujon 80n., 97n., 567n. ; B. 95. 

Beccaria xvi, 3n., 7n., lln. 



Boissevain B. 126. 

Bolles37n., 438n. 

Borght, R. V. d. 99n., 541; B. 55. 

Bourguin 7n., 8n., lOn., 12-13n., 
69n., 140n., 438n.; B. 132. 

Bourne 536, 541, 550, 567 ; B. 46-9. 

Bowley B. 123. 

British Association for the Advance- 
ment of Science, Reports of its 
Committee, 85n., 86n., 97n., 113n., 
123n., 207n., 495n. ; B. 99-102. 

Budelius 137n. 

Burchard 536 ; B. 53. 

Cairnes 27n., 438n., 439. 

Cantillon 132n. 

Carli 82, 220 ; his method 188, 193, 
203, 204, 383, 432, 534-6 ; B. 4. 

" Catallactics " 6n. 

Causes : not to be sought for in 
measurements 22-5, 26-7, 32, 483- 
4 ; an instance where wrongly 
used 37-8. 

Cibrario 79n. 

Clark, J. B., 8n. 

Classes of things having exchange- 
value 77. 

Clement, A. 439n. 

Coggeshall B. 91. 

Coincidence : of the averages or 
means, when the sums are con- 
stant 306-7, 318n. ; when the masses 
are constant 350-5, 510-14, 519-20, 
545n. ; of the three principal meth- 
ods, with two equally important 
classes 402-3; of still another with 
them 413. 

Commodity-standard 465. 

Compensation 47-8, 52-3 ; requires 
equality 233-4 ; arithmetic, har- 
monic, and geometric 238-9. 

Condillac 79n., 137n. 

Conrad 99n., 479n., 541 ; B. 96-7. 

Conservation of exchange- value 51, 
438-51. 



575 



576 



I^DEX 



Constancy of general exchange-value 
possible 44-53. 

Consumer, the average 85. 

Cost distinguished from price 124-5n. 

Cost-value : described 1-4 ; measure- 
ment of, often confounded with 
measurement of exchange-value 
24-5 ; one thing alone may rise or 
fall in 38-9 ; measured by quan- 
titv of labor required to produce 
the thing 57, 66n., 124, 126 n. ; 
we want all things to fall in 484, 
489. 

Courcelle-Seneuil 7n., 15n., 438 n. 

Cournot 38n., 66, 67, 69. 

Cross, W. 479n. 

Cunningham 208n. 

Dabos, H. lOn. 

Davenport, H. J. 123n. 

Del Mar 482n. 

Denis 10 n., 438n.; B. 125_. 

Deviation of the geometric average 
241-2, 313-24, 363-8, 517-519, 52l. 

Dick, G. H. 122n. 

Disappearance of old classes 113-14. 

Drobisch 41n., S4n., 86n., 97, 98, 99, 
111, 113n., 182, 194, 225, 387n., 540, 
541, 542 ; his method 194-201, 203, 
204, 205, 211, 261-2n., 383-6, 388, 
390, 392. 396, 427, 544, 547, 548, 
550, 55i; 559 ; B. 29-31. 

Dupre de Saint-Maur B. 3. 

Dutot 82, 122n., 220, 276; his method 
188, 203, 279n., 534, 554 ; B. 2. 

Earnings 127 ; see Wages. 

Economist, Tlae : its index-numbers 
83 ; its method 189, 432, 536, 561, 
565 ; see B. 21. 

Eden 85n. 

Edgeworth 77n., 85n., 86n., 122n., 
123n., 179n., ISOn., 203n., 207n., 
222, 223n., 224n. , 537, 538, 540, 
541, 542, 543, 544, 546, 548, 567, 
57 In.; B. 57-66. 

ElUs 537 ; B. 560. 

Ellissen, A. 31n. 

Elv, K. T. 438n., 439n. 

Engel 85n. 

Equality : in opposite variations 29- 
30 ; in compensatory variations, 
three ways of conceiving it 234ff., 
of distance travei-sed 236-7, 238, 
cf._ 246 ; of proportion 237, 238 ; 
arithmetic, harmonic, and geo- 
metric 237-8 ; indefinitely, the 
geometric 244. 

Error, measurement of, in the method 
for constant sums 332-41, in the 



universal methods 394, 402, 404-7, 
424-7. 

Esteem-value : described 1-6 ; meas- 
urement of, often confounded with 
the measurement of exchange-value 
24-5 ; one thing alone may per- 
haps rise or fall in 38-9 ; measured 
by esteem (and this by the quantity, 
or supply, of the thing) 58, 66n., 
also by the quantity of labor the 
thing Avill command 126n. ; com- 
pared with cost-value 124-5 ; 
measurement of, desirable 25, 125 ; 
measurement of, of commodities in 
general 125, 127-8 ; confusion in 
combining this measurement with 
the measurement of the exchange- 
value of monev 128-34 ; we want 
all things to fall in 484. 

Evelyn 82';84n., 122n., 132n., 208n.; 
his variety of Carli's method 188, 
432, 535, 536, 555, 556, 558 ; B. 5. 

Exchange-value : described 1-6 ; con- 
founded with other kinds of value 
4-6, 23-4, 38-9, 134, 484; rela- 
tively of 7-13, 55, this not peculiar 
56ft'. ; is a power 7, 12 ; a property 
q/'and mi things 7, 9-10 ; is not the 
other thing 11, nor the mere i-ela- 
tion with it 11 ; variability of 10, 
44-5, 480-1 ; is local and temporal 
14, 208n., 480; involves measure- 
ment 21-2 ; mea-surement of, to be 
kept distinct from that of other 
kinds of value 25, 133 ; particular 
10-13, 457 and n. ; individual 91 ; 
general 5n., 12-13n., 13 ; two kinds 
of general 13, 39-41, (this distinc- 
tion not peculiar 67-70), their con- 
sistency 43, their different behavior 
30, 31, 41-2, 48-51, relation be- 
tween their variations 454-7, 461-3, 
need of care to distinguish them 
468-71 ; one thing alone cannot 
change in 36-9 ; nor can all things 
rise or fall in, together 438-9 ; cf. 
441 ; emptiness of the desire that 
all things should fall in 484-7 ; 
mensuration of, like that of other 
simple attributes 64-6, with a pe- 
culiarity 70-1, how to meet this 
72-4 ; we may attain to a method 
of measuring, if not to a measure 
of 65 ; amount of precision needed 
74-5. 

Falkner 85n., 98, 438n., 536, 537; 

B. 111-13. 
Familv budgets 85n. 
Fauveau 113n.; 540 ; B. 54. 



indp:x 



577 



Fawcett 4, 37n. 

Ferrara, F. 4on. 

Fiamingo 25n. 

I'leetwood B. 1. 

Flux, A. W. 542n.; B. 140. 

Fonda 84n., 123n., 143n., 438n., 
479n., 493n., 537 ; B. 127. 

Forbes 222n. ; B. 78. 

Formulation : by equivalence 18, 
138, 171-2 ; by equality of ex- 
change-value 139 ff. ; by prices 173 
fi'.; reciprocal correspondence be- 
tween tliese 175-6 ; with uneven 
weighting 150-4, 178-80; with 
double weighting 154-6, 181-4, 
521-3, 544-52 ; using weighting 
not intended 159-69, 185-94, 535, 
537, 541, ignorantly introducing 
double weighting 193-4, 538-9, 
541-2; of exchange-value in all 
things 208-11, 444-6, 457-9. 

Foville, A. de 96n., 203n., 542, 567n. ; 
B. 50, 51. 

Fox well 85n., 207n., 567. 

Fraser, J. A. 534 ; B. 120. 

Frost, O. J. 479n. 

Galiani 45n., 137n. 

Garnet, L. A. 29n. 

Garnier 29n., 65-6n. 

Gauss 409 n., 424n. 

(ieometric average : behaves differ- 
ently from the geometric mean, see 
Deviation and Average ; studv of 
514 16. 

( ieometric mean : generally to be 
used in relations or ratios 105, 220; 
should be used in averaging weights 
105-10 ; general argument for 229- 
30, 231-2, 233ff. ; generally to be 
used in variations 244, 251-3, 254- 
5 ; see also Coincidence. 

Geyer 546n., 558; his method 536; 
B. 28. 

Gide 38n. 

Giffen84n.,96n., lOOn., 122n., 132n., 
478n., 554n.; liis method 537-8, 
567 and n. ; B. 38-45. 

Gold and Silver Commission 551. 

Grotius 478 and n. 

(iunton 475n. 

Hallam 13n. 

Hanauer 219 ; B. 35. 

Hansard 438n., 536 ; B. 67. 

Harris 127n. 

Harmonic average and mean : argu- 
ment for, in general 251 ; argument 
for, in averaging prices 25l)-7 (cf. 
215, 228-9), defect in this 260-2, 
37 



refutation of it when the sums are 
constant 294-6 ; study of 506-10. 

Hearn 6n. 

Hegel 353n. 

Heitz 97n. 

Held 45n. 

Helflerich, K. 25n. 

Holt, B. W. 128-9n. 

Horton 84n., 85n., 473,476, 478n. 

Hyde 122n. 

Inama-Sternegg, K. T. von 567n.; B. 
109. 

Index-numbers 83, 540 : careless re- 
jection of 189-90, 57 In. 

Individual, economic 89-94, 101, 
108-10, 118-20, 213n., 301n.; in- 
different in some methods 192n., 
307-9, 359, 387n. 

Institut international de Statistique, 
Reports of its Comite 567n. 

Intervening periods, the comparison 
should pass through 113n. 

Intrinsic: of qualities 7n. , 12n. ; of 
values 133n. 

James, H., his method 537, 542 ; B. 
10. 

Javolinus 137n. 

Jevons 6, lln., 22n , 36n, 38n., 41n., 
81n., 96n., 158, 178n., 180, 181n., 
184, 195, 220-1, 222n., 223, 224, 
232, 253, 257, 355, 383, 432, 435, 
477, 478n., 479n., 493n., 574 ; his 
dispute with Laspeyres 220-1, 224- 
5, 266-9. 275, 354 and n., 557, 
558, 559 ;' B. 22-4. 

Jourdain 438 n. 

Kant 353n. 
Knies 136n. 
Krai 534, 536 ; B. 98. 

Labor : not being exchangeable, has 
has not exchange-value 122-4 ; said 
to be the real price of things 4 and 
n. ; this wrong, it is the ultimati' 
cost of things 124-125 and n. 

Laspevres 98, 192n., 196n., 220, 221, 
222," 225, 383, 476, 477n., 546 ; the 
method he used 84n., 220, 536, 
545 ; the method he recommended 
99, 541, 550; his dispute witli 
Jevons, see Jevons ; B. 25-6. 

Laughlin 12n., 56n., 438n., 478n. 

Laves, T. 478 and n., 493n., 495n. 

Leber B. 12. 

Lehr 41n., 86n., Ill, 182, 192n., 197, 
198n., 207n., 222n., 225, 393, 476, 
477n., 544, 546, 571 ; his method 



578 



INDEX 



199, 204, 386-8, 391, 392n., 410, 

417, 418 and n., 419n., 420n., 422, 

423, 427, 428, 547-8 ; B. 68. 
Levasseur 41n., 82n., 137n., 438n., 

534, 542 ; B. 18. • 
Lexis 25n. 
Lindsay 41n., 113n., 122n., 541, 546, 

548.; B. 114. 
Locke 79. 
Logarithms : use of, in averaging 

180-1, 220. 
Lowe 84, 477, 478 and n., 493n.. 537 ; 

B. 8. 

McCuIloch7n.,24n.,45n.,122n.,137n. 

Maclaren 478n.; B. 17. 

Macleod 9n., lln., 13-14n., 29n., 
45n., 56n., 75n., 438n. 

Malthus 5n., lln., 24n., 81n.. 84n., 
126n., 127n., 135n., 486-7. 

Mannequin 57n., 137n., 438n., 439n., 
480n. 

Marshall 80n., 84n., 85n., 97n., 
113n., 123n., 222n., 478n., 495n., 
567 ; B. 93. 

Martello lOn., 23n., 45n., 303. 

Martin 80n., 122n., 208n., 567. 

Mass-unit : employment of the same 
in all classes 86, 90, 98, 164-5, 170, 
190, 545, 547; importance of 
rightly selecting 89-90, 156, 162, 
182, 183, 202, 269, 276, 285ff:, 
344ff., 550 ; indifferent in some 
cases 182n., 192n., 307-10, 359, 
377, 387n., 548. 

Mayo-Smith 208n., 222n.; B. 137-8. 

Mayr 567n. 

Mean : see Average ; median, and of 
greatest thickness 224. 

IMeasure of value : confusion in Mal- 
thus' s 24n. ; dispute concerning, 
suggested solution 126n. 

Measurement : of particular ex- 
change-values 14-15 ; involved in 
the conception of exchange-value 
21 ; independent of causes 22-5 ; of 
the exchange-value of money 137, 
how frequently to be made 493n. ; 
principles of, with constant sums 
305, with constant masses 359, for 
all cases 374-5. 

Measures : themselves to be measured 
57 and n., 137. 

Menger, C. 9-lOn. 

Mensuration, simple 58. 
Messedaglia 223-4, 257, 383 ; B. 52. 

Metrology 127. 

Mill, J. 122n. 

Mill, J. S. lln., 13n., 22n., 29n., 

37n., 41n., 438. 
Molinseus 137n. 



Money : its excliange-value the pri- 
mary one to be measured 20-1 ; de- 
nial of this rejected 137 ; question 
whetlier it is the standard of ex- 
change-value, cost-value, or esteem- 
value 24, 135, 488-9, 495; regula- 
tion of its quantity with a vieAv to 
keeping it steady in exchange-value 
489-93, or in cost-value or esteem- 
value 494-5. 

Money-value 5n. 

Montanari 137n. 

Mulhall 99n., 541 ; B. 74-75 

Multiple standard 79n., 473, 476, cf. 
478. 

Nasse96n., 122n.; B. 103-4. 
Natural : of price 4. 
Neumann-Spallart 567n. 
Newcomb 122n., 495n., 540; B. 76. 
Newmarch 536 ; B. 19-21. 
Nicholson 22n., 41n., 113n., 123n., 

143n., 225, 544, 567; his method 

197, 204, 391, 427n., 543, 544, 

548-51 ; B. 94. 
Nitti 23n. 
Nominal : of price 4 ; of value 4, 5 

and n., lln. 

Objective : two senses of 85-6. 
O' Conor, J. E. 219n. 
Oker, C. W. 190n. 
Osborne, G. P. 122n., 479n. 

Paasche 98, 99, 192n., 222n., 546 
and n.; his variety of Scrope's 
method 194, 197, 199, 204, 541; 
B. 33-4. 

Padan 190n., 192n., 222n., 540, 544; 
B. 141. 

Palgrave 83, 84n., lOOn., 219n.; his 
variety of Young's method 194, 197, 
199, 204, 433,537,538-9,567; B. 77. 

Pantaleoni 567 n. 

Parallelogram of forces : none in 
exchange-value 69n. 

Pareto 23n. 

Parsons 12n., 122n., 438n., 478n., 
479n., 493n., 540; B. 136. 

Patten 438n. 

Percentage : mathematically repre- 
sented in hundredths 28 ; three 
ways of reckoning 234 ff. 

Periods : question of, in weighting 
97-121 ; importance of, in the 
treatment of variations 238 ; gen- 
eral principles concerning 244, ap- 
plied 285 ff., 302n., 307, 343, 344, 
348, 353. 

Perry, A. L. 438n. 

Pierson, N. G. 190n., 222n.; B. 122 



INDEX. 



579 



"Pleasure-unit" 386, 393, 547. 

" Flutologv " 6n. 

Pollard, T' J. ]27n., 439n., 494n. 

Pomeroy 85n., 493n., 540; B. 135. 

Porter, G. R. 84n., 208n., 536; B. 11. 

Powers, L. G. 99n., 536, 541, 544n., 
547 ; B. 131. 

Preciousness : defined 165n.; measure 
of, of commodities 200-1, 211n. 

Price : variously used 4 and n. , 29 
and n. ; distinguished from cost 
124-5 n.; in formula?, see Formu- 
lation ; variations of, and variations 
of excliange-value 482-3 ; the de- 
sire that all prices should fall 488. 

Price, L. L., B. 138. 

Prince-Smith 35n., 3()n., 41n., 45n., 
136n., 143n. 

Prinsep, ('. (A 219n. 

Probability : wrongly invoked where 
measurement is possible 38-9, 61, 
69. 

Propositions: 114; 11 15 (33); III 
16; IV 16-17; V 17; VI 18 ; VII 
18 (113, 114, 139, 141, 470); VIII 
26 ; IX 28-9 (34n., 140, 447n. ); X 
29 (34n., 174); XI 29; XII 30 
(471); XIII 30 (140, 471); XIV 
31 (48, 213); XV 31 (447n. ); XVI 
32(447n. ; XVII 33 (42, 115n., 
149, 153, 196, 205n., 212, 311, 361, 
388); XVIII 34 (51, 212, 216n. ); 

XIX 34 (51, 212, 434, 455, 463); 

XX 36 (46, 150, 311, 3611; XXI 
41 (210); XXII 42 (210, 457); 
XXIII 42 (210); XXIV 42 (49, 
210, 45(3, 463); XXV 42 (115n., 
210, 211, 434, 45()n. ); XXVI 43 
(115n., 210, 457); XXVII 44 (115 
and n., 149, 153, 196, 205n., 310, 
361, 385, 388 ; XXVIII 47 (1-50, 
311, 361, 447n. ) ; XXIX 47 ( 447n. ) ; 
XXX 48 (91, 92n.); XXXI 48 
(434); XXXII 48 (50n., ^10, 
312n., 361n., 457); XXXIII 48-49 
(69n., 140, 214n., 446n., 453n., 
459n., 471); XXXIV 49 (453n.); 
XXXV 50 (210, 457); XXXVI 50 
(180, 210, 311-12, 361, 394, 396); 
XXXVII 51 ; XXXVIII 51 ; 
XXXIX 51 (438, 443, 446, 451); 
XL 52; XLI 114; XLII 114 
(440); XLIII 114 (440); XLIV 
115 (156, 179, 196, 205n., 310, 361, 
385, 388, 400); XLV 115 (156, 179, 
196, 205n., 311, 361, 388); XLVI 
213 ; XLVII 433 ; XLVIII 435 ; 
XLIX 436 ; L 437 ; LI 441 ; LII 
441 (485); LIII 447; LIV 449; 
LV 454; LVI 456; LVII 459 
(466, 471); LVIII 467. 



Pufendorf 476 and n. 
Purchase : meaning 173n. 
Purchasing power 5, 7, 18n., ll-12n., 
.30n., 31 n., 52, 172-3, 173n., .303-5. 

Quantitativeness of exchange-value 

14-22. 
Quantity : ambiquity of the term 8()n. 

Eeal : of price 4 ; of value 4, 5, 13n. ; 
of exchange-value 13. 

Relative: of value 3u., 5 ; no need 
of the epithet before exchange- 
value 39n.; all (juantities are 56, 
and all variations of quantities 6(>- 
7. 

Ricardo4-5, 7n., 23, 24n., 25n., 38n., 
84n., 126n., 135n., 137n., 439n 

Robertson, J. B., 479n. 

Rogers, J. E. T. 536 ; B. 92. 

Roscher 45, 122n., 225, 303, 546; 
B. 32. 

Rossi 45n., 136n. 

Sauerbeck 84n., 477n. ; his methods 
99n., 53(), 537,538, 541 ; B. 79-90. 

Sav, J. B., 5n., lln., 22n., 75n., 
79n. 

Schmid, F., B. 106. 

Scrope 220, 477, 478n., 495n.; his 
method, in general 191, 203, 280, 
350, 352, 353, 3(i0, 361, 303, 370, 

372, 382, 383, 396, 402, 408n., 
427-8, 430, 431, 443-5, 450, 534, 
539 44, 546n., 549, 556, emended, 

373, 37(>, 377, 395n., 396 to 407, 
applied to arithmetic means 99, 
376n., 377n., 409-10, 413-23, 
424n., 426n.; B. 9. 

Segnitz 97n. 

Senior, N. W. 438. 

Sergei, C. H. 534; B. 120. 

Series of periods : how to be formed 
206-7 ; inconsistency in cross- 
measurements 203 ( see Wester- 
gaard's test) ; special cases avoid- 
ing such inconsistency 204n., 390- 
2, 397-8 ; remedy, unsatisfactory 
334-6, 393, 398-9. 

Shadwell 12n., 27n., 127n., 494n. 

Sidgwick 84n., 99, 112, 113n., 540, 
544, 567 ; B. 56. 

Simon, A. 432 ; B. 72. 

Smith, Adam 4, 5, 8n., lln., 79n., 
122n., 124n., 126n., 137n. 

Smith, J. A. 479n., 540 ; B. 129. 

Smith, J. B., 556n. 

Smith, J. P. lOn., 208n., 536 ; B. 7. 

Soetbeer 536, 571 ; B. 14-16. 

Standard : sometimes outside 57 ; in 
simple mensuration nothing ulti- 



580 



INDEX 




V 



mate 58, itself a relation between a 
whole and its parts 58-9, 62, 63, 64, 
66, 70, in neither of them separately 
59-60 ; the whole only a practicable 
one 62 ; in exchange-value, should 
be as inclusive as possible 76-80. 

"Standard of desiderata" 85. 

Stewart, D. 79n. 

Storch 79n. 

Subjective : two senses 85-6. 

Taussig 85n.; B. 121. 

Test cases 324-41, 368-70, 423-33. 

" Timiotology " 6n. 

Todhunter 518n. 

Tooke 84n., 556n. 

Torrens 23n. 

Transposition : argument by 351-2 ; 

analysis of this 531-2. 
Trenholm 9n., lln. 
True-price 466-8, 473-7. 
Turgot 3, 4, 124n. ; 137n. 

Unit of exchange-value 136. 

Use-value : described 1-6 ; not de- 
sirable that things should fall in 
485u. 

Valeur : appreciative 3n. ; echangeable 
3 ; estimative 3. 

Value : defined 2-3 ; four kinds of 
1-6 ; these ought to be distin- 
guished by special epithets 1, 134 ; 
no other kinds of economic 128, 
but two more kinds 133n. ; faulty 
uses of the term and of allied terms 
36 and n. 

Variations : arithmetic, harmonic, 
and geometric 235-6 ; three kinds 
of, with reference to zero 245-8 ; 
compensatory 524-30 ; nature of 
geometric 531. 

Von Jacob 84 n. 

Wages : not to be counted in meas- 
uring variations in the exchange- 
value of money 121-33 ; wrong use 
of, alone, in measuring esteem- 
value 126-7, 130, 494. 

Walker, A. 534 ; B. 27. 

Walker, F. A. 8n., 56n., 137n., 478n. 

Walras, L. 6n., 7n., lOn., lln., 27n., 
36n., 45n., 96n., 138n., 178n., 
180n., 221-2, 223, 265n., 432, 479, 
480n., 493n., 540, 546n.; B. 69-72. 

Walsh, R. 479n. 

Walsh, R. H. 478n.; B. 13. 

Wasserab 84n., 122n., 537 ; B. 105. 

Weight : use of the term 81n. ; of the 
thing whose exchange-value in all 
otlier things is being measured, in- 



different 94-5 ; of classes varying 
like the average, indifferent 180n., 
500-1, cf. 115n.; of wages and 
of commodities, incommensurable 
132; of money 463-5. 

Weighting : defined 81 ; explained 
87-89 ; history of 84-7 ; haphazard 
81-2, 156, 170, 188, 201, 534; 
even 82, 501 ; proper uneven, need 
of 82-3 ; I'ough, preferable to even 
83, 121, 431-2 ; a wrong view of 
85 ; by mass-quantities 86-7, 90, 
97, 165, 170, 190, 195 ; by custom- 
house returns 96n. ; by consump- 
tion or production 95-6; according 
to relative sizes of the classes 81, 
501, these according to the num- 
bers of economic individuals in them 
89, 94, 120-1 ; question of periods, 
see under Periods ; uneven, in 
averaging the prices of each article 
during a period 96-7 ; even, in 
averaging the weights of periods 
105, 386 7 ; simple, alone possible 
in averaging variations 153 ; hidden 
506 ; perverted 161-6, 168, 186, 189, 
534, 535, 537 ; double 98, 111, 195, 
225, 521-3, 544-52, its need of prop- 
erly selected mass-units 156, 182, 
unintentionallv incurred 189, 193- 
4, 539, 541, ' 551 ; the weight- 
ing needed to make the geometric 
average good 239n., 318n., 408-9n. 

Wells, D. A. 24n. 

Westergaard 84n., 179n., 203, 206, 
222, 537 ; his test 205, 332n. , 370, 
389, 391n., 393, 396, 398, 399, 402, 
409, 417, 537, questioned 400-1; 
B. 110. 

Wetmore, W. S. 536, 570. 

Whateley 6n. 

Whewell 23n. 

Whitehead B. 119 

Whitelaw 479n., 493n., 536 ; B. 130. 

Wicksell99, 112, 123n., 222n., 332n., 
387n., 540, 544, 548; B. 139. 

Wiebe 122n., 536, 548 ; B. 124. 

Will, T. E. 479n. 

Williams, A. 479n., 493n. 

Wilson, W. D. 123n. 

Winn, H. 478n., 479n. 

Wright, C. D., B. 73. 

Young, Arthur 85, 97, 122n., 132n., 
536 ; his method 194, 203, 204, 
383, 428, 429, 432, 433, 536-9, 
539-40, 555 ; B. 6. 

Zuckerkandl 198n., 478n., 495n., 544, 
546, 548 ; B. 115-16. 



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